Theorem dfac8a 8853

Description: Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)

Assertion

Ref	Expression
dfac8a	|- ( A e. B -> ( E. h A. y e. ~P A ( y =/= (/) -> ( h ` y ) e. y ) -> A e. dom card ) )

Distinct variable groups:   y, h, A   B, h

Allowed substitution hint:   B( y)

Proof of Theorem dfac8a

Dummy variables f v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- recs ( ( v e. _V |-> ( h ` ( A \ ran v ) ) ) ) = recs ( ( v e. _V |-> ( h ` ( A \ ran v ) ) ) )
2		rneq 5351	. . . . 5 |- ( v = f -> ran v = ran f )
3	2	difeq2d 3728	. . . 4 |- ( v = f -> ( A \ ran v ) = ( A \ ran f ) )
4	3	fveq2d 6195	. . 3 |- ( v = f -> ( h ` ( A \ ran v ) ) = ( h ` ( A \ ran f ) ) )
5	4	cbvmptv 4750	. 2 |- ( v e. _V |-> ( h ` ( A \ ran v ) ) ) = ( f e. _V |-> ( h ` ( A \ ran f ) ) )
6	1, 5	dfac8alem 8852	1 |- ( A e. B -> ( E. h A. y e. ~P A ( y =/= (/) -> ( h ` y ) e. y ) -> A e. dom card ) )

Syntax hints:   -> wi 4   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ran crn 5115   ` cfv 5888  recscrecs 7467   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:  ween  8858  acnnum  8875  dfac8  8957