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Theorem dfac8alem 7900
Description: Lemma for dfac8a 7901. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
Hypotheses
Ref Expression
dfac8alem.2  |-  F  = recs ( G )
dfac8alem.3  |-  G  =  ( f  e.  _V  |->  ( g `  ( A  \  ran  f ) ) )
Assertion
Ref Expression
dfac8alem  |-  ( A  e.  C  ->  ( E. g A. y  e. 
~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
Distinct variable groups:    f, g,
y, A    C, g    f, F, y
Allowed substitution hints:    C( y, f)    F( g)    G( y, f, g)

Proof of Theorem dfac8alem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2956 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
2 difss 3466 . . . . . . . . . . . 12  |-  ( A 
\  ( F "
x ) )  C_  A
3 elpw2g 4355 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( A  \  ( F " x ) )  e.  ~P A  <->  ( A  \  ( F " x
) )  C_  A
) )
42, 3mpbiri 225 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( F "
x ) )  e. 
~P A )
5 neeq1 2606 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
6 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
g `  y )  =  ( g `  ( A  \  ( F " x ) ) ) )
7 id 20 . . . . . . . . . . . . . 14  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
86, 7eleq12d 2503 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( g `  y
)  e.  y  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
95, 8imbi12d 312 . . . . . . . . . . . 12  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  (
g `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
109rspcv 3040 . . . . . . . . . . 11  |-  ( ( A  \  ( F
" x ) )  e.  ~P A  -> 
( A. y  e. 
~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
114, 10syl 16 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
12113imp 1147 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A. y  e.  ~P  A
( y  =/=  (/)  ->  (
g `  y )  e.  y )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( g `
 ( A  \ 
( F " x
) ) )  e.  ( A  \  ( F " x ) ) )
13 dfac8alem.2 . . . . . . . . . . . 12  |-  F  = recs ( G )
1413tfr2 6651 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
1513tfr1 6650 . . . . . . . . . . . . . 14  |-  F  Fn  On
16 fnfun 5534 . . . . . . . . . . . . . 14  |-  ( F  Fn  On  ->  Fun  F )
1715, 16ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  F
18 vex 2951 . . . . . . . . . . . . 13  |-  x  e. 
_V
19 resfunexg 5949 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
2017, 18, 19mp2an 654 . . . . . . . . . . . 12  |-  ( F  |`  x )  e.  _V
21 rneq 5087 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
22 df-ima 4883 . . . . . . . . . . . . . . . 16  |-  ( F
" x )  =  ran  ( F  |`  x )
2321, 22syl6eqr 2485 . . . . . . . . . . . . . . 15  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
2423difeq2d 3457 . . . . . . . . . . . . . 14  |-  ( f  =  ( F  |`  x )  ->  ( A  \  ran  f )  =  ( A  \ 
( F " x
) ) )
2524fveq2d 5724 . . . . . . . . . . . . 13  |-  ( f  =  ( F  |`  x )  ->  (
g `  ( A  \  ran  f ) )  =  ( g `  ( A  \  ( F " x ) ) ) )
26 dfac8alem.3 . . . . . . . . . . . . 13  |-  G  =  ( f  e.  _V  |->  ( g `  ( A  \  ran  f ) ) )
27 fvex 5734 . . . . . . . . . . . . 13  |-  ( g `
 ( A  \ 
( F " x
) ) )  e. 
_V
2825, 26, 27fvmpt 5798 . . . . . . . . . . . 12  |-  ( ( F  |`  x )  e.  _V  ->  ( G `  ( F  |`  x
) )  =  ( g `  ( A 
\  ( F "
x ) ) ) )
2920, 28ax-mp 8 . . . . . . . . . . 11  |-  ( G `
 ( F  |`  x ) )  =  ( g `  ( A  \  ( F "
x ) ) )
3014, 29syl6eq 2483 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( F `  x )  =  ( g `  ( A  \  ( F " x ) ) ) )
3130eleq1d 2501 . . . . . . . . 9  |-  ( x  e.  On  ->  (
( F `  x
)  e.  ( A 
\  ( F "
x ) )  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3212, 31syl5ibrcom 214 . . . . . . . 8  |-  ( ( A  e.  _V  /\  A. y  e.  ~P  A
( y  =/=  (/)  ->  (
g `  y )  e.  y )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( x  e.  On  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
33323expia 1155 . . . . . . 7  |-  ( ( A  e.  _V  /\  A. y  e.  ~P  A
( y  =/=  (/)  ->  (
g `  y )  e.  y ) )  -> 
( ( A  \ 
( F " x
) )  =/=  (/)  ->  (
x  e.  On  ->  ( F `  x )  e.  ( A  \ 
( F " x
) ) ) ) )
3433com23 74 . . . . . 6  |-  ( ( A  e.  _V  /\  A. y  e.  ~P  A
( y  =/=  (/)  ->  (
g `  y )  e.  y ) )  -> 
( x  e.  On  ->  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) ) )
3534ralrimiv 2780 . . . . 5  |-  ( ( A  e.  _V  /\  A. y  e.  ~P  A
( y  =/=  (/)  ->  (
g `  y )  e.  y ) )  ->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
3635ex 424 . . . 4  |-  ( A  e.  _V  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A. x  e.  On  ( ( A 
\  ( F "
x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) ) )
3715tz7.49c 6695 . . . . . 6  |-  ( ( A  e.  _V  /\  A. x  e.  On  (
( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
3837ex 424 . . . . 5  |-  ( A  e.  _V  ->  ( A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A ) )
3918f1oen 7120 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  x  ~~  A )
40 isnumi 7823 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
4139, 40sylan2 461 . . . . . 6  |-  ( ( x  e.  On  /\  ( F  |`  x ) : x -1-1-onto-> A )  ->  A  e.  dom  card )
4241rexlimiva 2817 . . . . 5  |-  ( E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A  ->  A  e.  dom  card )
4338, 42syl6 31 . . . 4  |-  ( A  e.  _V  ->  ( A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) )  ->  A  e.  dom  card ) )
4436, 43syld 42 . . 3  |-  ( A  e.  _V  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
451, 44syl 16 . 2  |-  ( A  e.  C  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
4645exlimdv 1646 1  |-  ( A  e.  C  ->  ( E. g A. y  e. 
~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   class class class wbr 4204    e. cmpt 4258   Oncon0 4573   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -1-1-onto->wf1o 5445   ` cfv 5446  recscrecs 6624    ~~ cen 7098   cardccrd 7812
This theorem is referenced by:  dfac8a  7901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-en 7102  df-card 7816
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