Theorem infxpidmlem9 7503

Description: Lemma for infxpidm 7507. By Zorn's Lemma zorn 4769, the collection H (which we show here to be a set) has a maximal element.

Hypotheses

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

Assertion

Ref	Expression
infxpidmlem9	|- E.g e. H A.h e. H -. g (. h

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

Proof of Theorem infxpidmlem9

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . . . 5 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
2		unab 2257	. . . . 5 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}) = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3	1, 2	eqtr4 1490	. . . 4 |- H = ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)})
4		df-sn 2402	. . . . . 6 |- {(/)} = {f | f = (/)}
5		p0ex 2760	. . . . . 6 |- {(/)} e. V
6	4, 5	eqeltrr 1537	. . . . 5 |- {f | f = (/)} e. V
7		df-rex 1642	. . . . . . . 8 |- (E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t(t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
8		visset 1804	. . . . . . . . . . . 12 |- t e. V
9	8	elpw 2394	. . . . . . . . . . 11 |- (t e. P~A <-> t (_ A)
10	9	anbi1i 480	. . . . . . . . . 10 |- ((t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> (t (_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
11		ancom 435	. . . . . . . . . 10 |- ((t (_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t (_ A))
12		an23 484	. . . . . . . . . 10 |- (((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t (_ A) <-> ((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
13	10, 11, 12	3bitr 177	. . . . . . . . 9 |- ((t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
14	13	exbii 1047	. . . . . . . 8 |- (E.t(t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
15	7, 14	bitr 173	. . . . . . 7 |- (E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
16	15	abbii 1567	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} = {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}
17		infxpidmlem.2	. . . . . . . 8 |- A e. V
18	17	pwex 2735	. . . . . . 7 |- P~A e. V
19	8, 8	xpex 3250	. . . . . . . . 9 |- (t X. t) e. V
20		mapex 4312	. . . . . . . . 9 |- (((t X. t) e. V /\ t e. V) -> {f | f:(t X. t)-->t} e. V)
21	19, 8, 20	mp2an 695	. . . . . . . 8 |- {f | f:(t X. t)-->t} e. V
22		f1of 3674	. . . . . . . . . 10 |- (f:(t X. t)-1-1-onto->t -> f:(t X. t)-->t)
23	22	adantl 388	. . . . . . . . 9 |- ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) -> f:(t X. t)-->t)
24	23	ss2abi 2110	. . . . . . . 8 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} (_ {f | f:(t X. t)-->t}
25	21, 24	ssexi 2710	. . . . . . 7 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
26	18, 25	abrexex2 3856	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
27	16, 26	eqeltrr 1537	. . . . 5 |- {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)} e. V
28	6, 27	unex 2863	. . . 4 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}) e. V
29	3, 28	eqeltr 1536	. . 3 |- H e. V
30	29	zorn 4769	. 2 |- (A.z((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H) -> E.g e. H A.h e. H -. g (. h)
31		eqid 1468	. . 3 |- ran U. z = ran U. z
32		visset 1804	. . 3 |- z e. V
33	1, 31, 32	infxpidmlem8 7502	. 2 |- ((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H)
34	30, 33	mpg 983	1 |- E.g e. H A.h e. H -. g (. 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   u. cun 2035   (_ wss 2037   (. wpss 2038  (/)c0 2270  P~cpw 2391  {csn 2399  U.cuni 2493   class class class wbr 2609  omcom 3121   X. cxp 3158  ran crn 3161  -->wf 3168  -1-1-onto->wf1o 3171   ~<_ cdom 4349

This theorem is referenced by:  infxpidmlem12 7506

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  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  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-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  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-fv 3188  df-iso 3189  df-en 4351  df-dom 4352

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