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Theorem dfac8a 7913
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.
StepHypRef Expression
1 eqid 2438 . 2  |- recs ( ( v  e.  _V  |->  ( h `  ( A 
\  ran  v )
) ) )  = recs ( ( v  e. 
_V  |->  ( h `  ( A  \  ran  v
) ) ) )
2 rneq 5097 . . . . 5  |-  ( v  =  f  ->  ran  v  =  ran  f )
32difeq2d 3467 . . . 4  |-  ( v  =  f  ->  ( A  \  ran  v )  =  ( A  \  ran  f ) )
43fveq2d 5734 . . 3  |-  ( v  =  f  ->  (
h `  ( A  \  ran  v ) )  =  ( h `  ( A  \  ran  f
) ) )
54cbvmptv 4302 . 2  |-  ( v  e.  _V  |->  ( h `
 ( A  \  ran  v ) ) )  =  ( f  e. 
_V  |->  ( h `  ( A  \  ran  f
) ) )
61, 5dfac8alem 7912 1  |-  ( A  e.  B  ->  ( E. h A. y  e. 
~P  A ( y  =/=  (/)  ->  ( h `  y )  e.  y )  ->  A  e.  dom  card ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1551    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    \ cdif 3319   (/)c0 3630   ~Pcpw 3801    e. cmpt 4268   dom cdm 4880   ran crn 4881   ` cfv 5456  recscrecs 6634   cardccrd 7824
This theorem is referenced by:  ween  7918  acnnum  7935  dfac8  8017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-en 7112  df-card 7828
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