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Theorem dfac8a 7541
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
StepHypRef Expression
1 eqid 2253 . 2  |- recs ( ( v  e.  _V  |->  ( h `  ( A 
\  ran  v )
) ) )  = recs ( ( v  e. 
_V  |->  ( h `  ( A  \  ran  v
) ) ) )
2 rneq 4811 . . . . 5  |-  ( v  =  f  ->  ran  v  =  ran  f )
32difeq2d 3211 . . . 4  |-  ( v  =  f  ->  ( A  \  ran  v )  =  ( A  \  ran  f ) )
43fveq2d 5381 . . 3  |-  ( v  =  f  ->  (
h `  ( A  \  ran  v ) )  =  ( h `  ( A  \  ran  f
) ) )
54cbvmptv 4008 . 2  |-  ( v  e.  _V  |->  ( h `
 ( A  \  ran  v ) ) )  =  ( f  e. 
_V  |->  ( h `  ( A  \  ran  f
) ) )
61, 5dfac8alem 7540 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 6   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   _Vcvv 2727    \ cdif 3075   (/)c0 3362   ~Pcpw 3530    e. cmpt 3974   dom cdm 4580   ran crn 4581   ` cfv 4592  recscrecs 6273   cardccrd 7452
This theorem is referenced by:  ween  7546  acnnum  7563  dfac8  7645
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-en 6750  df-card 7456
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