Theorem infxpidmlem8 7502

Description: Lemma for infxpidm 7507. The union of a collection of chains C in the collection of bijections H belongs to H. This property will be needed to apply Zorn's Lemma in infxpidmlem9 7503.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem6.2	|- B = ran U. C
infxpidmlem8.3	|- C e. V

Assertion

Ref	Expression
infxpidmlem8	|- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C e. H)

Distinct variable groups:   f,g,h,t,A   B,f,g,h,t   C,f,g,h,t   g,H,h

Proof of Theorem infxpidmlem8

Step	Hyp	Ref	Expression
1		ssel2 2054	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> g e. H)
2		infxpidmlem.1	. . . . . . . . . . . . . 14 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3		visset 1804	. . . . . . . . . . . . . 14 |- g e. V
4	2, 3	infxpidmlem2 7496	. . . . . . . . . . . . 13 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
5	4	biimp 151	. . . . . . . . . . . 12 |- (g e. H -> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
6	5	ord 232	. . . . . . . . . . 11 |- (g e. H -> (-. g = (/) -> E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
7		f1ofo 3680	. . . . . . . . . . . . . . . . 17 |- (g:(x X. x)-1-1-onto->x -> g:(x X. x)-onto->x)
8		forn 3659	. . . . . . . . . . . . . . . . 17 |- (g:(x X. x)-onto->x -> ran g = x)
9	7, 8	syl 10	. . . . . . . . . . . . . . . 16 |- (g:(x X. x)-1-1-onto->x -> ran g = x)
10	9	eqcomd 1472	. . . . . . . . . . . . . . 15 |- (g:(x X. x)-1-1-onto->x -> x = ran g)
11	10	anim1i 334	. . . . . . . . . . . . . 14 |- ((g:(x X. x)-1-1-onto->x /\ (om ~<_ x /\ x (_ A)) -> (x = ran g /\ (om ~<_ x /\ x (_ A)))
12	11	ancoms 436	. . . . . . . . . . . . 13 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (x = ran g /\ (om ~<_ x /\ x (_ A)))
13	12	19.22i 1036	. . . . . . . . . . . 12 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> E.x(x = ran g /\ (om ~<_ x /\ x (_ A)))
14		rnexg 3345	. . . . . . . . . . . . . 14 |- (g e. V -> ran g e. V)
15	3, 14	ax-mp 7	. . . . . . . . . . . . 13 |- ran g e. V
16		breq2 2613	. . . . . . . . . . . . . 14 |- (x = ran g -> (om ~<_ x <-> om ~<_ ran g))
17		sseq1 2072	. . . . . . . . . . . . . 14 |- (x = ran g -> (x (_ A <-> ran g (_ A))
18	16, 17	anbi12d 626	. . . . . . . . . . . . 13 |- (x = ran g -> ((om ~<_ x /\ x (_ A) <-> (om ~<_ ran g /\ ran g (_ A)))
19	15, 18	ceqsexv 1826	. . . . . . . . . . . 12 |- (E.x(x = ran g /\ (om ~<_ x /\ x (_ A)) <-> (om ~<_ ran g /\ ran g (_ A))
20	13, 19	sylib 198	. . . . . . . . . . 11 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ ran g /\ ran g (_ A))
21	6, 20	syl6 22	. . . . . . . . . 10 |- (g e. H -> (-. g = (/) -> (om ~<_ ran g /\ ran g (_ A)))
22	1, 21	syl 10	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> (om ~<_ ran g /\ ran g (_ A)))
23		domtr 4396	. . . . . . . . . . . . 13 |- ((om ~<_ ran g /\ ran g ~<_ B) -> om ~<_ B)
24		ra4e 1687	. . . . . . . . . . . . . . . . 17 |- ((g e. C /\ y e. ran g) -> E.g e. C y e. ran g)
25		infxpidmlem6.2	. . . . . . . . . . . . . . . . . 18 |- B = ran U. C
26	2, 25	infxpidmlem6 7500	. . . . . . . . . . . . . . . . 17 |- (y e. B <-> E.g e. C y e. ran g)
27	24, 26	sylibr 200	. . . . . . . . . . . . . . . 16 |- ((g e. C /\ y e. ran g) -> y e. B)
28	27	ex 373	. . . . . . . . . . . . . . 15 |- (g e. C -> (y e. ran g -> y e. B))
29	28	ssrdv 2060	. . . . . . . . . . . . . 14 |- (g e. C -> ran g (_ B)
30		ssdomg 4389	. . . . . . . . . . . . . . 15 |- (ran g e. V -> (ran g (_ B -> ran g ~<_ B))
31	15, 30	ax-mp 7	. . . . . . . . . . . . . 14 |- (ran g (_ B -> ran g ~<_ B)
32	29, 31	syl 10	. . . . . . . . . . . . 13 |- (g e. C -> ran g ~<_ B)
33	23, 32	sylan2 451	. . . . . . . . . . . 12 |- ((om ~<_ ran g /\ g e. C) -> om ~<_ B)
34	33	expcom 374	. . . . . . . . . . 11 |- (g e. C -> (om ~<_ ran g -> om ~<_ B))
35	34	adantl 388	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> (om ~<_ ran g -> om ~<_ B))
36	35	adantrd 391	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> ((om ~<_ ran g /\ ran g (_ A) -> om ~<_ B))
37	22, 36	syld 27	. . . . . . . 8 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> om ~<_ B))
38	37	r19.23adva 1739	. . . . . . 7 |- (C (_ H -> (E.g e. C -. g = (/) -> om ~<_ B))
39		uni0b 2513	. . . . . . . . . 10 |- (U.C = (/) <-> C (_ {(/)})
40		dfss3 2049	. . . . . . . . . 10 |- (C (_ {(/)} <-> A.g e. C g e. {(/)})
41		elsn 2411	. . . . . . . . . . 11 |- (g e. {(/)} <-> g = (/))
42	41	ralbii 1659	. . . . . . . . . 10 |- (A.g e. C g e. {(/)} <-> A.g e. C g = (/))
43	39, 40, 42	3bitr 177	. . . . . . . . 9 |- (U.C = (/) <-> A.g e. C g = (/))
44	43	negbii 187	. . . . . . . 8 |- (-. U.C = (/) <-> -. A.g e. C g = (/))
45		rexnal 1646	. . . . . . . 8 |- (E.g e. C -. g = (/) <-> -. A.g e. C g = (/))
46	44, 45	bitr4 176	. . . . . . 7 |- (-. U.C = (/) <-> E.g e. C -. g = (/))
47	38, 46	syl5ib 206	. . . . . 6 |- (C (_ H -> (-. U.C = (/) -> om ~<_ B))
48		pm3.27 323	. . . . . . . . . . . 12 |- ((om ~<_ ran g /\ ran g (_ A) -> ran g (_ A)
49	22, 48	syl6 22	. . . . . . . . . . 11 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> ran g (_ A))
50		rneq 3328	. . . . . . . . . . . . 13 |- (g = (/) -> ran g = ran (/))
51		rn0 3341	. . . . . . . . . . . . 13 |- ran (/) = (/)
52	50, 51	syl6eq 1515	. . . . . . . . . . . 12 |- (g = (/) -> ran g = (/))
53		0ss 2291	. . . . . . . . . . . . 13 |- (/) (_ A
54	53	a1i 8	. . . . . . . . . . . 12 |- (g = (/) -> (/) (_ A)
55	52, 54	eqsstrd 2085	. . . . . . . . . . 11 |- (g = (/) -> ran g (_ A)
56	49, 55	pm2.61d2 129	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> ran g (_ A)
57	56	sseld 2057	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> (y e. ran g -> y e. A))
58	57	r19.23adva 1739	. . . . . . . 8 |- (C (_ H -> (E.g e. C y e. ran g -> y e. A))
59	58, 26	syl5ib 206	. . . . . . 7 |- (C (_ H -> (y e. B -> y e. A))
60	59	ssrdv 2060	. . . . . 6 |- (C (_ H -> B (_ A)
61	47, 60	jctird 600	. . . . 5 |- (C (_ H -> (-. U.C = (/) -> (om ~<_ B /\ B (_ A)))
62	61	adantr 389	. . . 4 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> (om ~<_ B /\ B (_ A)))
63	2, 25	infxpidmlem7 7501	. . . 4 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C:(B X. B)-1-1-onto->B)
64	62, 63	jctird 600	. . 3 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> ((om ~<_ B /\ B (_ A) /\ U.C:(B X. B)-1-1-onto->B)))
65		infxpidmlem8.3	. . . . 5 |- C e. V
66	65	uniex 2861	. . . 4 |- U.C e. V
67		rnexg 3345	. . . . . 6 |- (U.C e. V -> ran U. C e. V)
68	66, 67	ax-mp 7	. . . . 5 |- ran U. C e. V
69	25, 68	eqeltr 1536	. . . 4 |- B e. V
70	2, 66, 69	infxpidmlem3 7497	. . 3 |- (((om ~<_ B /\ B (_ A) /\ U.C:(B X. B)-1-1-onto->B) -> U.C e. H)
71	64, 70	syl6 22	. 2 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> U.C e. H))
72		orc 269	. . 3 |- (U.C = (/) -> (U.C = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ U.C:(x X. x)-1-1-onto->x)))
73	2, 66	infxpidmlem2 7496	. . 3 |- (U.C e. H <-> (U.C = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ U.C:(x X. x)-1-1-onto->x)))
74	72, 73	sylibr 200	. 2 |- (U.C = (/) -> U.C e. H)
75	71, 74	pm2.61d2 129	1 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C e. H)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  A.wral 1637  E.wrex 1638  Vcvv 1802   (_ wss 2037  (/)c0 2270  {csn 2399  U.cuni 2493   class class class wbr 2609  omcom 3121   X. cxp 3158  ran crn 3161  -onto->wfo 3170  -1-1-onto->wf1o 3171   ~<_ cdom 4349

This theorem is referenced by:  infxpidmlem9 7503

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-en 4351  df-dom 4352

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