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Theorem dfac8alem 7672
Description: Lemma for dfac8a 7673. 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 2809 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
2 difss 3316 . . . . . . . . . . . 12  |-  ( A 
\  ( F "
x ) )  C_  A
3 elpw2g 4190 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( A  \  ( F " x ) )  e.  ~P A  <->  ( A  \  ( F " x
) )  C_  A
) )
42, 3mpbiri 224 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( F "
x ) )  e. 
~P A )
5 neeq1 2467 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
6 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
g `  y )  =  ( g `  ( A  \  ( F " x ) ) ) )
7 id 19 . . . . . . . . . . . . . 14  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
86, 7eleq12d 2364 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( g `  y
)  e.  y  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
95, 8imbi12d 311 . . . . . . . . . . . 12  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  (
g `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
109rspcv 2893 . . . . . . . . . . 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 15 . . . . . . . . . 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 1145 . . . . . . . . 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 6430 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
1513tfr1 6429 . . . . . . . . . . . . . 14  |-  F  Fn  On
16 fnfun 5357 . . . . . . . . . . . . . 14  |-  ( F  Fn  On  ->  Fun  F )
1715, 16ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  F
18 vex 2804 . . . . . . . . . . . . 13  |-  x  e. 
_V
19 resfunexg 5753 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
2017, 18, 19mp2an 653 . . . . . . . . . . . 12  |-  ( F  |`  x )  e.  _V
21 rneq 4920 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
22 df-ima 4718 . . . . . . . . . . . . . . . 16  |-  ( F
" x )  =  ran  ( F  |`  x )
2321, 22syl6eqr 2346 . . . . . . . . . . . . . . 15  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
2423difeq2d 3307 . . . . . . . . . . . . . 14  |-  ( f  =  ( F  |`  x )  ->  ( A  \  ran  f )  =  ( A  \ 
( F " x
) ) )
2524fveq2d 5545 . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . 13  |-  ( g `
 ( A  \ 
( F " x
) ) )  e. 
_V
2825, 26, 27fvmpt 5618 . . . . . . . . . . . 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 2344 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( F `  x )  =  ( g `  ( A  \  ( F " x ) ) ) )
3130eleq1d 2362 . . . . . . . . 9  |-  ( x  e.  On  ->  (
( F `  x
)  e.  ( A 
\  ( F "
x ) )  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3212, 31syl5ibrcom 213 . . . . . . . 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 1153 . . . . . . 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 72 . . . . . 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 2638 . . . . 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 423 . . . 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 6474 . . . . . 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 423 . . . . 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 6898 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  x  ~~  A )
40 isnumi 7595 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
4139, 40sylan2 460 . . . . . 6  |-  ( ( x  e.  On  /\  ( F  |`  x ) : x -1-1-onto-> A )  ->  A  e.  dom  card )
4241rexlimiva 2675 . . . . 5  |-  ( E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A  ->  A  e.  dom  card )
4338, 42syl6 29 . . . 4  |-  ( A  e.  _V  ->  ( A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) )  ->  A  e.  dom  card ) )
4436, 43syld 40 . . 3  |-  ( A  e.  _V  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
451, 44syl 15 . 2  |-  ( A  e.  C  ->  ( A. y  e.  ~P  A ( y  =/=  (/)  ->  ( g `  y )  e.  y )  ->  A  e.  dom  card ) )
4645exlimdv 1626 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 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   Oncon0 4408   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271  recscrecs 6403    ~~ cen 6876   cardccrd 7584
This theorem is referenced by:  dfac8a  7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-en 6880  df-card 7588
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