Theorem canthnumlem 8927

Description: Lemma for canthnum 8928. (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 5715	. . . . 5 |- ( F : ( ~P A i^i dom card ) -1-1-> A -> F : ( ~P A i^i dom card ) --> A )
2		ssid 3484	. . . . . 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 8926	. . . . . 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 1304	. . . . 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 474	. . . 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 1002	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( F ` B ) = ( F ` C ) )
10		simpr 461	. . . 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 1000	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B C_ A )
12		elpw2g 4564	. . . . . . 7 |- ( A e. V -> ( B e. ~P A <-> B C_ A ) )
13	12	adantr 465	. . . . . 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 232	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. ~P A )
15		eqid 2454	. . . . . . . . . . . . 13 |- B = B
16		eqid 2454	. . . . . . . . . . . . 13 |- ( W ` B ) = ( W ` B )
17	15, 16	pm3.2i 455	. . . . . . . . . . . 12 |- ( B = B /\ ( W ` B ) = ( W ` B ) )
18		elex 3087	. . . . . . . . . . . . . 14 |- ( A e. V -> A e. _V )
19	18	adantr 465	. . . . . . . . . . . . 13 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> A e. _V )
20	10, 1	syl 16	. . . . . . . . . . . . . 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 5953	. . . . . . . . . . . . 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 8925	. . . . . . . . . . . 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 233	. . . . . . . . . . 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 459	. . . . . . . . . 10 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B W ( W ` B ) )
25	3, 19	fpwwelem 8924	. . . . . . . . . 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 210	. . . . . . . . 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 463	. . . . . . . 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 459	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( W ` B ) We B )
29		fvex 5810	. . . . . . . 8 |- ( W ` B ) e. _V
30		weeq1 4817	. . . . . . . 8 |- ( r = ( W ` B ) -> ( r We B <-> ( W ` B ) We B ) )
31	29, 30	spcev 3170	. . . . . . 7 |- ( ( W ` B ) We B -> E. r r We B )
32	28, 31	syl 16	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> E. r r We B )
33		ween 8317	. . . . . 6 |- ( B e. dom card <-> E. r r We B )
34	32, 33	sylibr 212	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. dom card )
35	14, 34	elind 3649	. . . 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 1001	. . . . . . . 8 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C. B )
37	36	pssssd 3562	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ B )
38	37, 11	sstrd 3475	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ A )
39		elpw2g 4564	. . . . . . 7 |- ( A e. V -> ( C e. ~P A <-> C C_ A ) )
40	39	adantr 465	. . . . . 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 232	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ~P A )
42		ssnum 8321	. . . . . 6 |- ( ( B e. dom card /\ C C_ B ) -> C e. dom card )
43	34, 37, 42	syl2anc 661	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. dom card )
44	41, 43	elind 3649	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ( ~P A i^i dom card ) )
45		f1fveq 6085	. . . 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 1217	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
47	9, 46	mpbid 210	. 2 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B = C )
48	36	pssned 3563	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C =/= B )
49	48	necomd 2723	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B =/= C )
50	49	neneqd 2655	. 2 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> -. B = C )
51	47, 50	pm2.65da 576	1 |- ( A e. V -> -. F : ( ~P A i^i dom card ) -1-1-> A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   E.wex 1587   e. wcel 1758   A.wral 2799   _Vcvv 3078   i^i cin 3436   C_ wss 3437   C. wpss 3438   ~Pcpw 3969   {csn 3986   U.cuni 4200   class class class wbr 4401   {copab 4458   We wwe 4787   X. cxp 4947   `'ccnv 4948   dom cdm 4949   "cima 4952   -->wf 5523   -1-1->wf1 5524   ` cfv 5527   cardccrd 8217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-1st 6688  df-recs 6943  df-er 7212  df-en 7422  df-dom 7423  df-oi 7836  df-card 8221

This theorem is referenced by:  canthnum  8928