Theorem canthnumlem 9070

Description: Lemma for canthnum 9071. (Contributed by Mario Carneiro, 19-May-2015.)

Hypotheses

Ref	Expression
canth4.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
canth4.2	|- B = U. dom W
canth4.3	|- C = ( `' ( W ` B ) " { ( F ` B ) } )

Assertion

Ref	Expression
canthnumlem	|- ( A e. V -> -. F : ( ~P A i^i dom card ) -1-1-> A )

Distinct variable groups:   x, r, y, A   B, r, x, y   F, r, x, y   V, r, x, y   y, C   W, r, x, y

Allowed substitution hints:   C( x, r)

Proof of Theorem canthnumlem

Step	Hyp	Ref	Expression
1		f1f 5777	. . . . 5 |- ( F : ( ~P A i^i dom card ) -1-1-> A -> F : ( ~P A i^i dom card ) --> A )
2		ssid 3450	. . . . . 6 |- ( ~P A i^i dom card ) C_ ( ~P A i^i dom card )
3		canth4.1	. . . . . . 7 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
4		canth4.2	. . . . . . 7 |- B = U. dom W
5		canth4.3	. . . . . . 7 |- C = ( `' ( W ` B ) " { ( F ` B ) } )
6	3, 4, 5	canth4 9069	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) --> A /\ ( ~P A i^i dom card ) C_ ( ~P A i^i dom card ) ) -> ( B C_ A /\ C C. B /\ ( F ` B ) = ( F ` C ) ) )
7	2, 6	mp3an3 1352	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) --> A ) -> ( B C_ A /\ C C. B /\ ( F ` B ) = ( F ` C ) ) )
8	1, 7	sylan2 477	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( B C_ A /\ C C. B /\ ( F ` B ) = ( F ` C ) ) )
9	8	simp3d 1021	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( F ` B ) = ( F ` C ) )
10		simpr 463	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> F : ( ~P A i^i dom card ) -1-1-> A )
11	8	simp1d 1019	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B C_ A )
12		elpw2g 4565	. . . . . . 7 |- ( A e. V -> ( B e. ~P A <-> B C_ A ) )
13	12	adantr 467	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( B e. ~P A <-> B C_ A ) )
14	11, 13	mpbird 236	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. ~P A )
15		eqid 2450	. . . . . . . . . . . . 13 |- B = B
16		eqid 2450	. . . . . . . . . . . . 13 |- ( W ` B ) = ( W ` B )
17	15, 16	pm3.2i 457	. . . . . . . . . . . 12 |- ( B = B /\ ( W ` B ) = ( W ` B ) )
18		elex 3053	. . . . . . . . . . . . . 14 |- ( A e. V -> A e. _V )
19	18	adantr 467	. . . . . . . . . . . . 13 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> A e. _V )
20	10, 1	syl 17	. . . . . . . . . . . . . 14 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> F : ( ~P A i^i dom card ) --> A )
21	20	ffvelrnda 6020	. . . . . . . . . . . . 13 |- ( ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) /\ x e. ( ~P A i^i dom card ) ) -> ( F ` x ) e. A )
22	3, 19, 21, 4	fpwwe 9068	. . . . . . . . . . . 12 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( ( B W ( W ` B ) /\ ( F ` B ) e. B ) <-> ( B = B /\ ( W ` B ) = ( W ` B ) ) ) )
23	17, 22	mpbiri 237	. . . . . . . . . . 11 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( B W ( W ` B ) /\ ( F ` B ) e. B ) )
24	23	simpld 461	. . . . . . . . . 10 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B W ( W ` B ) )
25	3, 19	fpwwelem 9067	. . . . . . . . . 10 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( B W ( W ` B ) <-> ( ( B C_ A /\ ( W ` B ) C_ ( B X. B ) ) /\ ( ( W ` B ) We B /\ A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y ) ) ) )
26	24, 25	mpbid 214	. . . . . . . . 9 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( ( B C_ A /\ ( W ` B ) C_ ( B X. B ) ) /\ ( ( W ` B ) We B /\ A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y ) ) )
27	26	simprd 465	. . . . . . . 8 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( ( W ` B ) We B /\ A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y ) )
28	27	simpld 461	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( W ` B ) We B )
29		fvex 5873	. . . . . . . 8 |- ( W ` B ) e. _V
30		weeq1 4821	. . . . . . . 8 |- ( r = ( W ` B ) -> ( r We B <-> ( W ` B ) We B ) )
31	29, 30	spcev 3140	. . . . . . 7 |- ( ( W ` B ) We B -> E. r r We B )
32	28, 31	syl 17	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> E. r r We B )
33		ween 8463	. . . . . 6 |- ( B e. dom card <-> E. r r We B )
34	32, 33	sylibr 216	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. dom card )
35	14, 34	elind 3617	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. ( ~P A i^i dom card ) )
36	8	simp2d 1020	. . . . . . . 8 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C. B )
37	36	pssssd 3529	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ B )
38	37, 11	sstrd 3441	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ A )
39		elpw2g 4565	. . . . . . 7 |- ( A e. V -> ( C e. ~P A <-> C C_ A ) )
40	39	adantr 467	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( C e. ~P A <-> C C_ A ) )
41	38, 40	mpbird 236	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ~P A )
42		ssnum 8467	. . . . . 6 |- ( ( B e. dom card /\ C C_ B ) -> C e. dom card )
43	34, 37, 42	syl2anc 666	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. dom card )
44	41, 43	elind 3617	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ( ~P A i^i dom card ) )
45		f1fveq 6161	. . . 4 |- ( ( F : ( ~P A i^i dom card ) -1-1-> A /\ ( B e. ( ~P A i^i dom card ) /\ C e. ( ~P A i^i dom card ) ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
46	10, 35, 44, 45	syl12anc 1265	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
47	9, 46	mpbid 214	. 2 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B = C )
48	36	pssned 3530	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C =/= B )
49	48	necomd 2678	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B =/= C )
50	49	neneqd 2628	. 2 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> -. B = C )
51	47, 50	pm2.65da 579	1 |- ( A e. V -> -. F : ( ~P A i^i dom card ) -1-1-> A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   E.wex 1662   e. wcel 1886   A.wral 2736   _Vcvv 3044   i^i cin 3402   C_ wss 3403   C. wpss 3404   ~Pcpw 3950   {csn 3967   U.cuni 4197   class class class wbr 4401   {copab 4459   We wwe 4791   X. cxp 4831   `'ccnv 4832   dom cdm 4833   "cima 4836   -->wf 5577   -1-1->wf1 5578   ` cfv 5581   cardccrd 8366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-1st 6790  df-wrecs 7025  df-recs 7087  df-er 7360  df-en 7567  df-dom 7568  df-oi 8022  df-card 8370

This theorem is referenced by:  canthnum  9071