Theorem infmap2lem2 7580

Description: Lemma for infmap2 7581. Given the relation R, we use the Axiom of Choice ac7g 4749 to extract a function f that provides the one-to-one mapping for the dominance relation.

Hypotheses

Ref	Expression
infmap2lem.1	|- A e. V
infmap2lem.2	|- B e. V
infmap2lem.3	|- R = {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)}

Assertion

Ref	Expression
infmap2lem2	|- {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)

Distinct variable groups:   x,z,w,A   x,B,z,w

Proof of Theorem infmap2lem2

Step	Hyp	Ref	Expression
1		infmap2lem.3	. . . 4 |- R = {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)}
2		df-xp 3184	. . . . . 6 |- (P~A X. (A ^m B)) = {<.z, w>. | (z e. P~A /\ w e. (A ^m B))}
3		infmap2lem.1	. . . . . . . 8 |- A e. V
4	3	pwex 2745	. . . . . . 7 |- P~A e. V
5		oprex 3983	. . . . . . 7 |- (A ^m B) e. V
6	4, 5	xpex 3260	. . . . . 6 |- (P~A X. (A ^m B)) e. V
7	2, 6	eqeltrr 1545	. . . . 5 |- {<.z, w>. | (z e. P~A /\ w e. (A ^m B))} e. V
8		pm3.26 319	. . . . . . . . 9 |- ((z (_ A /\ w:B-onto->z) -> z (_ A)
9		visset 1813	. . . . . . . . . 10 |- z e. V
10	9	elpw 2404	. . . . . . . . 9 |- (z e. P~A <-> z (_ A)
11	8, 10	sylibr 200	. . . . . . . 8 |- ((z (_ A /\ w:B-onto->z) -> z e. P~A)
12		fof 3672	. . . . . . . . . . . 12 |- (w:B-onto->z -> w:B-->z)
13		ffn 3627	. . . . . . . . . . . 12 |- (w:B-->z -> w Fn B)
14	12, 13	syl 10	. . . . . . . . . . 11 |- (w:B-onto->z -> w Fn B)
15	14	adantl 388	. . . . . . . . . 10 |- ((z (_ A /\ w:B-onto->z) -> w Fn B)
16		forn 3674	. . . . . . . . . . . 12 |- (w:B-onto->z -> ran w = z)
17	16	sseq1d 2088	. . . . . . . . . . 11 |- (w:B-onto->z -> (ran w (_ A <-> z (_ A))
18	17	biimparc 419	. . . . . . . . . 10 |- ((z (_ A /\ w:B-onto->z) -> ran w (_ A)
19	15, 18	jca 288	. . . . . . . . 9 |- ((z (_ A /\ w:B-onto->z) -> (w Fn B /\ ran w (_ A))
20		infmap2lem.2	. . . . . . . . . . 11 |- B e. V
21	3, 20	elmap 4334	. . . . . . . . . 10 |- (w e. (A ^m B) <-> w:B-->A)
22		df-f 3194	. . . . . . . . . 10 |- (w:B-->A <-> (w Fn B /\ ran w (_ A))
23	21, 22	bitr 173	. . . . . . . . 9 |- (w e. (A ^m B) <-> (w Fn B /\ ran w (_ A))
24	19, 23	sylibr 200	. . . . . . . 8 |- ((z (_ A /\ w:B-onto->z) -> w e. (A ^m B))
25	11, 24	jca 288	. . . . . . 7 |- ((z (_ A /\ w:B-onto->z) -> (z e. P~A /\ w e. (A ^m B)))
26	25	adantlr 393	. . . . . 6 |- (((z (_ A /\ z ~~ B) /\ w:B-onto->z) -> (z e. P~A /\ w e. (A ^m B)))
27	26	ssopab2i 2823	. . . . 5 |- {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)} (_ {<.z, w>. | (z e. P~A /\ w e. (A ^m B))}
28	7, 27	ssexi 2720	. . . 4 |- {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)} e. V
29	1, 28	eqeltr 1544	. . 3 |- R e. V
30		ac7g 4749	. . 3 |- (R e. V -> E.f(f (_ R /\ f Fn dom R))
31	29, 30	ax-mp 7	. 2 |- E.f(f (_ R /\ f Fn dom R)
32		df-pw 2402	. . . . . 6 |- P~A = {x | x (_ A}
33	32, 4	eqeltrr 1545	. . . . 5 |- {x | x (_ A} e. V
34		pm3.26 319	. . . . . 6 |- ((x (_ A /\ x ~~ B) -> x (_ A)
35	34	ss2abi 2120	. . . . 5 |- {x | (x (_ A /\ x ~~ B)} (_ {x | x (_ A}
36	33, 35	ssexi 2720	. . . 4 |- {x | (x (_ A /\ x ~~ B)} e. V
37	3, 20, 1	infmap2lem1 7579	. . . . . 6 |- ((f (_ R /\ f Fn dom R) -> (v e. {x | (x (_ A /\ x ~~ B)} -> (v (_ A /\ (f` v):B-onto->v)))
38		fss 3635	. . . . . . . . 9 |- (((f` v):B-->v /\ v (_ A) -> (f` v):B-->A)
39		fof 3672	. . . . . . . . 9 |- ((f` v):B-onto->v -> (f` v):B-->v)
40	38, 39	sylan 448	. . . . . . . 8 |- (((f` v):B-onto->v /\ v (_ A) -> (f` v):B-->A)
41	40	ancoms 436	. . . . . . 7 |- ((v (_ A /\ (f` v):B-onto->v) -> (f` v):B-->A)
42	3, 20	elmap 4334	. . . . . . 7 |- ((f` v) e. (A ^m B) <-> (f` v):B-->A)
43	41, 42	sylibr 200	. . . . . 6 |- ((v (_ A /\ (f` v):B-onto->v) -> (f` v) e. (A ^m B))
44	37, 43	syl6 22	. . . . 5 |- ((f (_ R /\ f Fn dom R) -> (v e. {x | (x (_ A /\ x ~~ B)} -> (f` v) e. (A ^m B)))
45		pm3.27 323	. . . . . . . 8 |- ((v (_ A /\ (f` v):B-onto->v) -> (f` v):B-onto->v)
46	37, 45	syl6 22	. . . . . . 7 |- ((f (_ R /\ f Fn dom R) -> (v e. {x | (x (_ A /\ x ~~ B)} -> (f` v):B-onto->v))
47	3, 20, 1	infmap2lem1 7579	. . . . . . . 8 |- ((f (_ R /\ f Fn dom R) -> (u e. {x | (x (_ A /\ x ~~ B)} -> (u (_ A /\ (f` u):B-onto->u)))
48		pm3.27 323	. . . . . . . 8 |- ((u (_ A /\ (f` u):B-onto->u) -> (f` u):B-onto->u)
49	47, 48	syl6 22	. . . . . . 7 |- ((f (_ R /\ f Fn dom R) -> (u e. {x | (x (_ A /\ x ~~ B)} -> (f` u):B-onto->u))
50	46, 49	anim12d 558	. . . . . 6 |- ((f (_ R /\ f Fn dom R) -> ((v e. {x | (x (_ A /\ x ~~ B)} /\ u e. {x | (x (_ A /\ x ~~ B)}) -> ((f` v):B-onto->v /\ (f` u):B-onto->u)))
51		forn 3674	. . . . . . . . 9 |- ((f` v):B-onto->v -> ran ( f` v) = v)
52		forn 3674	. . . . . . . . 9 |- ((f` u):B-onto->u -> ran ( f` u) = u)
53	51, 52	eqeqan12d 1490	. . . . . . . 8 |- (((f` v):B-onto->v /\ (f` u):B-onto->u) -> (ran ( f` v) = ran ( f` u) <-> v = u))
54		rneq 3339	. . . . . . . 8 |- ((f` v) = (f` u) -> ran ( f` v) = ran ( f` u))
55	53, 54	syl5bi 208	. . . . . . 7 |- (((f` v):B-onto->v /\ (f` u):B-onto->u) -> ((f` v) = (f` u) -> v = u))
56		fveq2 3724	. . . . . . 7 |- (v = u -> (f` v) = (f` u))
57	55, 56	impbid1 517	. . . . . 6 |- (((f` v):B-onto->v /\ (f` u):B-onto->u) -> ((f` v) = (f` u) <-> v = u))
58	50, 57	syl6 22	. . . . 5 |- ((f (_ R /\ f Fn dom R) -> ((v e. {x | (x (_ A /\ x ~~ B)} /\ u e. {x | (x (_ A /\ x ~~ B)}) -> ((f` v) = (f` u) <-> v = u)))
59	44, 58	dom2d 4404	. . . 4 |- ((f (_ R /\ f Fn dom R) -> ({x | (x (_ A /\ x ~~ B)} e. V -> {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)))
60	36, 59	mpi 44	. . 3 |- ((f (_ R /\ f Fn dom R) -> {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B))
61	60	19.23aiv 1295	. 2 |- (E.f(f (_ R /\ f Fn dom R) -> {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B))
62	31, 61	ax-mp 7	1 |- {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811   (_ wss 2047  P~cpw 2401   class class class wbr 2619  {copab 2666   X. cxp 3168  dom cdm 3170  ran crn 3171   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963   ^m cm 4322   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  infmap2 7581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-er 4261  df-map 4324  df-en 4368  df-dom 4369

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