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Theorem dfac8alem 8442
Description: Lemma for dfac8a 8443. 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 3068 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
2 difss 3570 . . . . . . . . . . . 12  |-  ( A 
\  ( F "
x ) )  C_  A
3 elpw2g 4557 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( A  \  ( F " x ) )  e.  ~P A  <->  ( A  \  ( F " x
) )  C_  A
) )
42, 3mpbiri 233 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( F "
x ) )  e. 
~P A )
5 neeq1 2684 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
6 fveq2 5849 . . . . . . . . . . . . . 14  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
g `  y )  =  ( g `  ( A  \  ( F " x ) ) ) )
7 id 22 . . . . . . . . . . . . . 14  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
86, 7eleq12d 2484 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( g `  y
)  e.  y  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
95, 8imbi12d 318 . . . . . . . . . . . 12  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  (
g `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
109rspcv 3156 . . . . . . . . . . 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 17 . . . . . . . . . 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 1191 . . . . . . . . 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 7101 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
1513tfr1 7100 . . . . . . . . . . . . . 14  |-  F  Fn  On
16 fnfun 5659 . . . . . . . . . . . . . 14  |-  ( F  Fn  On  ->  Fun  F )
1715, 16ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  F
18 vex 3062 . . . . . . . . . . . . 13  |-  x  e. 
_V
19 resfunexg 6118 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
2017, 18, 19mp2an 670 . . . . . . . . . . . 12  |-  ( F  |`  x )  e.  _V
21 rneq 5049 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
22 df-ima 4836 . . . . . . . . . . . . . . . 16  |-  ( F
" x )  =  ran  ( F  |`  x )
2321, 22syl6eqr 2461 . . . . . . . . . . . . . . 15  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
2423difeq2d 3561 . . . . . . . . . . . . . 14  |-  ( f  =  ( F  |`  x )  ->  ( A  \  ran  f )  =  ( A  \ 
( F " x
) ) )
2524fveq2d 5853 . . . . . . . . . . . . 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 5859 . . . . . . . . . . . . 13  |-  ( g `
 ( A  \ 
( F " x
) ) )  e. 
_V
2825, 26, 27fvmpt 5932 . . . . . . . . . . . 12  |-  ( ( F  |`  x )  e.  _V  ->  ( G `  ( F  |`  x
) )  =  ( g `  ( A 
\  ( F "
x ) ) ) )
2920, 28ax-mp 5 . . . . . . . . . . 11  |-  ( G `
 ( F  |`  x ) )  =  ( g `  ( A  \  ( F "
x ) ) )
3014, 29syl6eq 2459 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( F `  x )  =  ( g `  ( A  \  ( F " x ) ) ) )
3130eleq1d 2471 . . . . . . . . 9  |-  ( x  e.  On  ->  (
( F `  x
)  e.  ( A 
\  ( F "
x ) )  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3212, 31syl5ibrcom 222 . . . . . . . 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 1199 . . . . . . 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 78 . . . . . 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 2816 . . . . 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 432 . . . 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 7148 . . . . . 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 432 . . . . 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 7574 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  x  ~~  A )
40 isnumi 8359 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
4139, 40sylan2 472 . . . . . 6  |-  ( ( x  e.  On  /\  ( F  |`  x ) : x -1-1-onto-> A )  ->  A  e.  dom  card )
4241rexlimiva 2892 . . . . 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 17 . 2  |-  ( A  e.  C  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
4645exlimdv 1745 1  |-  ( A  e.  C  ->  ( E. g A. y  e. 
~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   _Vcvv 3059    \ cdif 3411    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   class class class wbr 4395    |-> cmpt 4453   dom cdm 4823   ran crn 4824    |` cres 4825   "cima 4826   Oncon0 5410   Fun wfun 5563    Fn wfn 5564   -1-1-onto->wf1o 5568   ` cfv 5569  recscrecs 7074    ~~ cen 7551   cardccrd 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-wrecs 7013  df-recs 7075  df-en 7555  df-card 8352
This theorem is referenced by:  dfac8a  8443
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