Theorem dfac8alem 8852

Description: Lemma for dfac8a 8853. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)

Hypotheses

Ref	Expression
dfac8alem.2	|- F = recs ( G )
dfac8alem.3	|- G = ( f e. _V |-> ( g ` ( A \ ran f ) ) )

Assertion

Ref	Expression
dfac8alem	|- ( A e. C -> ( E. g A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> A e. dom card ) )

Distinct variable groups:   f, g, y, A   C, g   f, F, y

Allowed substitution hints:   C( y, f)   F( g)   G( y, f, g)

Proof of Theorem dfac8alem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( A e. C -> A e. _V )
2		difss 3737	. . . . . . . . . . . 12 |- ( A \ ( F " x ) ) C_ A
3		elpw2g 4827	. . . . . . . . . . . 12 |- ( A e. _V -> ( ( A \ ( F " x ) ) e. ~P A <-> ( A \ ( F " x ) ) C_ A ) )
4	2, 3	mpbiri 248	. . . . . . . . . . 11 |- ( A e. _V -> ( A \ ( F " x ) ) e. ~P A )
5		neeq1 2856	. . . . . . . . . . . . 13 |- ( y = ( A \ ( F " x ) ) -> ( y =/= (/) <-> ( A \ ( F " x ) ) =/= (/) ) )
6		fveq2 6191	. . . . . . . . . . . . . 14 |- ( y = ( A \ ( F " x ) ) -> ( g ` y ) = ( g ` ( A \ ( F " x ) ) ) )
7		id 22	. . . . . . . . . . . . . 14 |- ( y = ( A \ ( F " x ) ) -> y = ( A \ ( F " x ) ) )
8	6, 7	eleq12d 2695	. . . . . . . . . . . . 13 |- ( y = ( A \ ( F " x ) ) -> ( ( g ` y ) e. y <-> ( g ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) )
9	5, 8	imbi12d 334	. . . . . . . . . . . 12 |- ( y = ( A \ ( F " x ) ) -> ( ( y =/= (/) -> ( g ` y ) e. y ) <-> ( ( A \ ( F " x ) ) =/= (/) -> ( g ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) ) )
10	9	rspcv 3305	. . . . . . . . . . 11 |- ( ( A \ ( F " x ) ) e. ~P A -> ( A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> ( ( A \ ( F " x ) ) =/= (/) -> ( g ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) ) )
11	4, 10	syl 17	. . . . . . . . . 10 |- ( A e. _V -> ( A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> ( ( A \ ( F " x ) ) =/= (/) -> ( g ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) ) )
12	11	3imp 1256	. . . . . . . . 9 |- ( ( A e. _V /\ A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) /\ ( A \ ( F " x ) ) =/= (/) ) -> ( g ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) )
13		dfac8alem.2	. . . . . . . . . . . 12 |- F = recs ( G )
14	13	tfr2 7494	. . . . . . . . . . 11 |- ( x e. On -> ( F ` x ) = ( G ` ( F |` x ) ) )
15	13	tfr1 7493	. . . . . . . . . . . . . 14 |- F Fn On
16		fnfun 5988	. . . . . . . . . . . . . 14 |- ( F Fn On -> Fun F )
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 |- Fun F
18		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
19		resfunexg 6479	. . . . . . . . . . . . 13 |- ( ( Fun F /\ x e. _V ) -> ( F |` x ) e. _V )
20	17, 18, 19	mp2an 708	. . . . . . . . . . . 12 |- ( F |` x ) e. _V
21		rneq 5351	. . . . . . . . . . . . . . . 16 |- ( f = ( F |` x ) -> ran f = ran ( F |` x ) )
22		df-ima 5127	. . . . . . . . . . . . . . . 16 |- ( F " x ) = ran ( F |` x )
23	21, 22	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( f = ( F |` x ) -> ran f = ( F " x ) )
24	23	difeq2d 3728	. . . . . . . . . . . . . 14 |- ( f = ( F |` x ) -> ( A \ ran f ) = ( A \ ( F " x ) ) )
25	24	fveq2d 6195	. . . . . . . . . . . . 13 |- ( f = ( F |` x ) -> ( g ` ( A \ ran f ) ) = ( g ` ( A \ ( F " x ) ) ) )
26		dfac8alem.3	. . . . . . . . . . . . 13 |- G = ( f e. _V |-> ( g ` ( A \ ran f ) ) )
27		fvex 6201	. . . . . . . . . . . . 13 |- ( g ` ( A \ ( F " x ) ) ) e. _V
28	25, 26, 27	fvmpt 6282	. . . . . . . . . . . 12 |- ( ( F |` x ) e. _V -> ( G ` ( F |` x ) ) = ( g ` ( A \ ( F " x ) ) ) )
29	20, 28	ax-mp 5	. . . . . . . . . . 11 |- ( G ` ( F |` x ) ) = ( g ` ( A \ ( F " x ) ) )
30	14, 29	syl6eq 2672	. . . . . . . . . 10 |- ( x e. On -> ( F ` x ) = ( g ` ( A \ ( F " x ) ) ) )
31	30	eleq1d 2686	. . . . . . . . 9 |- ( x e. On -> ( ( F ` x ) e. ( A \ ( F " x ) ) <-> ( g ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) )
32	12, 31	syl5ibrcom 237	. . . . . . . 8 |- ( ( A e. _V /\ A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) /\ ( A \ ( F " x ) ) =/= (/) ) -> ( x e. On -> ( F ` x ) e. ( A \ ( F " x ) ) ) )
33	32	3expia 1267	. . . . . . 7 |- ( ( A e. _V /\ A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) ) -> ( ( A \ ( F " x ) ) =/= (/) -> ( x e. On -> ( F ` x ) e. ( A \ ( F " x ) ) ) ) )
34	33	com23 86	. . . . . 6 |- ( ( A e. _V /\ A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) ) -> ( x e. On -> ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) ) )
35	34	ralrimiv 2965	. . . . 5 |- ( ( A e. _V /\ A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) ) -> A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) )
36	35	ex 450	. . . 4 |- ( A e. _V -> ( A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) ) )
37	15	tz7.49c 7541	. . . . . 6 |- ( ( A e. _V /\ A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) ) -> E. x e. On ( F |` x ) : x -1-1-onto-> A )
38	37	ex 450	. . . . 5 |- ( A e. _V -> ( A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) -> E. x e. On ( F |` x ) : x -1-1-onto-> A ) )
39	18	f1oen 7976	. . . . . . 7 |- ( ( F |` x ) : x -1-1-onto-> A -> x ~~ A )
40		isnumi 8772	. . . . . . 7 |- ( ( x e. On /\ x ~~ A ) -> A e. dom card )
41	39, 40	sylan2 491	. . . . . 6 |- ( ( x e. On /\ ( F |` x ) : x -1-1-onto-> A ) -> A e. dom card )
42	41	rexlimiva 3028	. . . . 5 |- ( E. x e. On ( F |` x ) : x -1-1-onto-> A -> A e. dom card )
43	38, 42	syl6 35	. . . 4 |- ( A e. _V -> ( A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) -> A e. dom card ) )
44	36, 43	syld 47	. . 3 |- ( A e. _V -> ( A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> A e. dom card ) )
45	1, 44	syl 17	. 2 |- ( A e. C -> ( A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> A e. dom card ) )
46	45	exlimdv 1861	1 |- ( A e. C -> ( E. g A. y e. ~P A ( y =/= (/) -> ( g ` y ) e. y ) -> A e. dom card ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  recscrecs 7467   ~~ cen 7952   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468  df-en 7956  df-card 8765

This theorem is referenced by:  dfac8a  8853