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Theorem dfac8a 7653
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 2284 . 2  |- recs ( ( v  e.  _V  |->  ( h `  ( A 
\  ran  v )
) ) )  = recs ( ( v  e. 
_V  |->  ( h `  ( A  \  ran  v
) ) ) )
2 rneq 4903 . . . . 5  |-  ( v  =  f  ->  ran  v  =  ran  f )
32difeq2d 3295 . . . 4  |-  ( v  =  f  ->  ( A  \  ran  v )  =  ( A  \  ran  f ) )
43fveq2d 5490 . . 3  |-  ( v  =  f  ->  (
h `  ( A  \  ran  v ) )  =  ( h `  ( A  \  ran  f
) ) )
54cbvmptv 4112 . 2  |-  ( v  e.  _V  |->  ( h `
 ( A  \  ran  v ) ) )  =  ( f  e. 
_V  |->  ( h `  ( A  \  ran  f
) ) )
61, 5dfac8alem 7652 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 1528    = wceq 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   _Vcvv 2789    \ cdif 3150   (/)c0 3456   ~Pcpw 3626    e. cmpt 4078    dom cdm 4688   ran crn 4689   ` cfv 5221  recscrecs 6383   cardccrd 7564
This theorem is referenced by:  ween  7658  acnnum  7675  dfac8  7757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-en 6860  df-card 7568
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