Theorem canthnumlem 9022

Description: Lemma for canthnum 9023. (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 5779	. . . . 5 |- ( F : ( ~P A i^i dom card ) -1-1-> A -> F : ( ~P A i^i dom card ) --> A )
2		ssid 3523	. . . . . 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 9021	. . . . . 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 1313	. . . . 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 1010	. . 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 1008	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B C_ A )
12		elpw2g 4610	. . . . . . 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 2467	. . . . . . . . . . . . 13 |- B = B
16		eqid 2467	. . . . . . . . . . . . 13 |- ( W ` B ) = ( W ` B )
17	15, 16	pm3.2i 455	. . . . . . . . . . . 12 |- ( B = B /\ ( W ` B ) = ( W ` B ) )
18		elex 3122	. . . . . . . . . . . . . 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 6019	. . . . . . . . . . . . 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 9020	. . . . . . . . . . . 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 9019	. . . . . . . . . 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 5874	. . . . . . . 8 |- ( W ` B ) e. _V
30		weeq1 4867	. . . . . . . 8 |- ( r = ( W ` B ) -> ( r We B <-> ( W ` B ) We B ) )
31	29, 30	spcev 3205	. . . . . . 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 8412	. . . . . 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 3688	. . . 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 1009	. . . . . . . 8 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C. B )
37	36	pssssd 3601	. . . . . . 7 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ B )
38	37, 11	sstrd 3514	. . . . . 6 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C C_ A )
39		elpw2g 4610	. . . . . . 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 8416	. . . . . 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 3688	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C e. ( ~P A i^i dom card ) )
45		f1fveq 6156	. . . 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 1226	. . 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 3602	. . . 4 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> C =/= B )
49	48	necomd 2738	. . 3 |- ( ( A e. V /\ F : ( ~P A i^i dom card ) -1-1-> A ) -> B =/= C )
50	49	neneqd 2669	. 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 973   = wceq 1379   E.wex 1596   e. wcel 1767   A.wral 2814   _Vcvv 3113   i^i cin 3475   C_ wss 3476   C. wpss 3477   ~Pcpw 4010   {csn 4027   U.cuni 4245   class class class wbr 4447   {copab 4504   We wwe 4837   X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   -->wf 5582   -1-1->wf1 5583   ` cfv 5586   cardccrd 8312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-1st 6781  df-recs 7039  df-er 7308  df-en 7514  df-dom 7515  df-oi 7931  df-card 8316

This theorem is referenced by:  canthnum  9023