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Theorem dfac8alem 8406
Description: Lemma for dfac8a 8407. 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 3122 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
2 difss 3631 . . . . . . . . . . . 12  |-  ( A 
\  ( F "
x ) )  C_  A
3 elpw2g 4610 . . . . . . . . . . . 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 2748 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
6 fveq2 5864 . . . . . . . . . . . . . 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 2549 . . . . . . . . . . . . 13  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( g `  y
)  e.  y  <->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
95, 8imbi12d 320 . . . . . . . . . . . 12  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  (
g `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( g `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
109rspcv 3210 . . . . . . . . . . 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 1190 . . . . . . . . 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 7064 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
1513tfr1 7063 . . . . . . . . . . . . . 14  |-  F  Fn  On
16 fnfun 5676 . . . . . . . . . . . . . 14  |-  ( F  Fn  On  ->  Fun  F )
1715, 16ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  F
18 vex 3116 . . . . . . . . . . . . 13  |-  x  e. 
_V
19 resfunexg 6124 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
2017, 18, 19mp2an 672 . . . . . . . . . . . 12  |-  ( F  |`  x )  e.  _V
21 rneq 5226 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
22 df-ima 5012 . . . . . . . . . . . . . . . 16  |-  ( F
" x )  =  ran  ( F  |`  x )
2321, 22syl6eqr 2526 . . . . . . . . . . . . . . 15  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
2423difeq2d 3622 . . . . . . . . . . . . . 14  |-  ( f  =  ( F  |`  x )  ->  ( A  \  ran  f )  =  ( A  \ 
( F " x
) ) )
2524fveq2d 5868 . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . 13  |-  ( g `
 ( A  \ 
( F " x
) ) )  e. 
_V
2825, 26, 27fvmpt 5948 . . . . . . . . . . . 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 2524 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( F `  x )  =  ( g `  ( A  \  ( F " x ) ) ) )
3130eleq1d 2536 . . . . . . . . 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 1198 . . . . . . 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 2876 . . . . 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 434 . . . 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 7108 . . . . . 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 434 . . . . 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 7533 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  x  ~~  A )
40 isnumi 8323 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
4139, 40sylan2 474 . . . . . 6  |-  ( ( x  e.  On  /\  ( F  |`  x ) : x -1-1-onto-> A )  ->  A  e.  dom  card )
4241rexlimiva 2951 . . . . 5  |-  ( E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A  ->  A  e.  dom  card )
4338, 42syl6 33 . . . 4  |-  ( A  e.  _V  ->  ( A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) )  ->  A  e.  dom  card ) )
4436, 43syld 44 . . 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 1700 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 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5580    Fn wfn 5581   -1-1-onto->wf1o 5585   ` cfv 5586  recscrecs 7038    ~~ cen 7510   cardccrd 8312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-recs 7039  df-en 7514  df-card 8316
This theorem is referenced by:  dfac8a  8407
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