Theorem canthnumlem 9091

Description: Lemma for canthnum 9092. (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 5792	. . . . 5 |- ( F : ( ~P A i^i dom card ) -1-1-> A -> F : ( ~P A i^i dom card ) --> A )
2		ssid 3437	. . . . . 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 9090	. . . . . 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 1379	. . . . 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 482	. . . 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 1044	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( F ` B ) = ( F ` C ) )
10		simpr 468	. . . 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 1042	. . . . . 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 472	. . . . . 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 240	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. ~P A )
15		eqid 2471	. . . . . . . . . . . . 13 |- B = B
16		eqid 2471	. . . . . . . . . . . . 13 |- ( W ` B ) = ( W ` B )
17	15, 16	pm3.2i 462	. . . . . . . . . . . 12 |- ( B = B /\ ( W ` B ) = ( W ` B ) )
18		elex 3040	. . . . . . . . . . . . . 14 |- ( A e. V -> A e. _V )
19	18	adantr 472	. . . . . . . . . . . . 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 6037	. . . . . . . . . . . . 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 9089	. . . . . . . . . . . 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 241	. . . . . . . . . . 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 466	. . . . . . . . . 10 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B W ( W ` B ) )
25	3, 19	fpwwelem 9088	. . . . . . . . . 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 215	. . . . . . . . 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 470	. . . . . . . 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 466	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( W ` B ) We B )
29		fvex 5889	. . . . . . . 8 |- ( W ` B ) e. _V
30		weeq1 4827	. . . . . . . 8 |- ( r = ( W ` B ) -> ( r We B <-> ( W ` B ) We B ) )
31	29, 30	spcev 3127	. . . . . . 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 8484	. . . . . 6 |- ( B e. dom card <-> E. r r We B )
34	32, 33	sylibr 217	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B e. dom card )
35	14, 34	elind 3609	. . . 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 1043	. . . . . . . 8 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C. B )
37	36	pssssd 3516	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ B )
38	37, 11	sstrd 3428	. . . . . 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 472	. . . . . 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 240	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ~P A )
42		ssnum 8488	. . . . . 6 |- ( ( B e. dom card /\ C C_ B ) -> C e. dom card )
43	34, 37, 42	syl2anc 673	. . . . 5 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. dom card )
44	41, 43	elind 3609	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ( ~P A i^i dom card ) )
45		f1fveq 6181	. . . 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 1290	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
47	9, 46	mpbid 215	. 2 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B = C )
48	36	pssned 3517	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C =/= B )
49	48	necomd 2698	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B =/= C )
50	49	neneqd 2648	. 2 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> -. B = C )
51	47, 50	pm2.65da 586	1 |- ( A e. V -> -. F : ( ~P A i^i dom card ) -1-1-> A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   A.wral 2756   _Vcvv 3031   i^i cin 3389   C_ wss 3390   C. wpss 3391   ~Pcpw 3942   {csn 3959   U.cuni 4190   class class class wbr 4395   {copab 4453   We wwe 4797   X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842   -->wf 5585   -1-1->wf1 5586   ` cfv 5589   cardccrd 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-1st 6812  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-oi 8043  df-card 8391

This theorem is referenced by:  canthnum  9092