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Theorem dfac8a 7659
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 2285 . 2  |- recs ( ( v  e.  _V  |->  ( h `  ( A 
\  ran  v )
) ) )  = recs ( ( v  e. 
_V  |->  ( h `  ( A  \  ran  v
) ) ) )
2 rneq 4906 . . . . 5  |-  ( v  =  f  ->  ran  v  =  ran  f )
32difeq2d 3296 . . . 4  |-  ( v  =  f  ->  ( A  \  ran  v )  =  ( A  \  ran  f ) )
43fveq2d 5531 . . 3  |-  ( v  =  f  ->  (
h `  ( A  \  ran  v ) )  =  ( h `  ( A  \  ran  f
) ) )
54cbvmptv 4113 . 2  |-  ( v  e.  _V  |->  ( h `
 ( A  \  ran  v ) ) )  =  ( f  e. 
_V  |->  ( h `  ( A  \  ran  f
) ) )
61, 5dfac8alem 7658 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 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   _Vcvv 2790    \ cdif 3151   (/)c0 3457   ~Pcpw 3627    e. cmpt 4079   dom cdm 4691   ran crn 4692   ` cfv 5257  recscrecs 6389   cardccrd 7570
This theorem is referenced by:  ween  7664  acnnum  7681  dfac8  7763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-en 6866  df-card 7574
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