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Theorem dfac8a 7804
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 2366 . 2  |- recs ( ( v  e.  _V  |->  ( h `  ( A 
\  ran  v )
) ) )  = recs ( ( v  e. 
_V  |->  ( h `  ( A  \  ran  v
) ) ) )
2 rneq 5007 . . . . 5  |-  ( v  =  f  ->  ran  v  =  ran  f )
32difeq2d 3381 . . . 4  |-  ( v  =  f  ->  ( A  \  ran  v )  =  ( A  \  ran  f ) )
43fveq2d 5636 . . 3  |-  ( v  =  f  ->  (
h `  ( A  \  ran  v ) )  =  ( h `  ( A  \  ran  f
) ) )
54cbvmptv 4213 . 2  |-  ( v  e.  _V  |->  ( h `
 ( A  \  ran  v ) ) )  =  ( f  e. 
_V  |->  ( h `  ( A  \  ran  f
) ) )
61, 5dfac8alem 7803 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 1546    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   _Vcvv 2873    \ cdif 3235   (/)c0 3543   ~Pcpw 3714    e. cmpt 4179   dom cdm 4792   ran crn 4793   ` cfv 5358  recscrecs 6529   cardccrd 7715
This theorem is referenced by:  ween  7809  acnnum  7826  dfac8  7908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-suc 4501  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-recs 6530  df-en 7007  df-card 7719
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