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Theorem dnnumch1 27141
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7657 (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
dnnumch.a  |-  ( ph  ->  A  e.  V )
dnnumch.g  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
Assertion
Ref Expression
dnnumch1  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Distinct variable groups:    x, F, y    x, G, y, z   
x, A, y, z    ph, x
Allowed substitution hints:    ph( y, z)    F( z)    V( x, y, z)

Proof of Theorem dnnumch1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dnnumch.a . 2  |-  ( ph  ->  A  e.  V )
2 recsval 6417 . . . . . . 7  |-  ( x  e.  On  ->  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) ) `
 x )  =  ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) `  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x ) ) )
3 dnnumch.f . . . . . . . 8  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
43fveq1i 5526 . . . . . . 7  |-  ( F `
 x )  =  (recs ( ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) ) ) `  x )
53tfr1 6413 . . . . . . . . . . 11  |-  F  Fn  On
6 fnfun 5341 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Fun  F )
75, 6ax-mp 8 . . . . . . . . . 10  |-  Fun  F
8 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
9 resfunexg 5737 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
107, 8, 9mp2an 653 . . . . . . . . 9  |-  ( F  |`  x )  e.  _V
11 rneq 4904 . . . . . . . . . . . . 13  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ran  ( F  |`  x ) )
12 df-ima 4702 . . . . . . . . . . . . 13  |-  ( F
" x )  =  ran  ( F  |`  x )
1311, 12syl6eqr 2333 . . . . . . . . . . . 12  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ( F " x ) )
1413difeq2d 3294 . . . . . . . . . . 11  |-  ( w  =  ( F  |`  x )  ->  ( A  \  ran  w )  =  ( A  \ 
( F " x
) ) )
1514fveq2d 5529 . . . . . . . . . 10  |-  ( w  =  ( F  |`  x )  ->  ( G `  ( A  \  ran  w ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
16 rneq 4904 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  ran  z  =  ran  w )
1716difeq2d 3294 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( A  \  ran  z )  =  ( A  \  ran  w ) )
1817fveq2d 5529 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( G `  ( A  \  ran  z ) )  =  ( G `  ( A  \  ran  w
) ) )
1918cbvmptv 4111 . . . . . . . . . 10  |-  ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) )  =  ( w  e. 
_V  |->  ( G `  ( A  \  ran  w
) ) )
20 fvex 5539 . . . . . . . . . 10  |-  ( G `
 ( A  \ 
( F " x
) ) )  e. 
_V
2115, 19, 20fvmpt 5602 . . . . . . . . 9  |-  ( ( F  |`  x )  e.  _V  ->  ( (
z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) `  ( F  |`  x ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
2210, 21ax-mp 8 . . . . . . . 8  |-  ( ( z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) `  ( F  |`  x ) )  =  ( G `  ( A  \  ( F " x ) ) )
233reseq1i 4951 . . . . . . . . 9  |-  ( F  |`  x )  =  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x )
2423fveq2i 5528 . . . . . . . 8  |-  ( ( z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) `  ( F  |`  x ) )  =  ( ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) ) `
 (recs ( ( z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) )  |`  x ) )
2522, 24eqtr3i 2305 . . . . . . 7  |-  ( G `
 ( A  \ 
( F " x
) ) )  =  ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) `  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x ) )
262, 4, 253eqtr4g 2340 . . . . . 6  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( A  \  ( F " x ) ) ) )
2726ad2antlr 707 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  =  ( G `  ( A  \  ( F "
x ) ) ) )
28 difss 3303 . . . . . . . . 9  |-  ( A 
\  ( F "
x ) )  C_  A
29 elpw2g 4174 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( A  \  ( F " x ) )  e.  ~P A  <->  ( A  \  ( F " x
) )  C_  A
) )
301, 29syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  e.  ~P A 
<->  ( A  \  ( F " x ) ) 
C_  A ) )
3128, 30mpbiri 224 . . . . . . . 8  |-  ( ph  ->  ( A  \  ( F " x ) )  e.  ~P A )
32 dnnumch.g . . . . . . . 8  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
33 neeq1 2454 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
34 fveq2 5525 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  ( G `  y )  =  ( G `  ( A  \  ( F " x ) ) ) )
35 id 19 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
3634, 35eleq12d 2351 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( G `  y
)  e.  y  <->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3733, 36imbi12d 311 . . . . . . . . 9  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  ( G `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
3837rspcva 2882 . . . . . . . 8  |-  ( ( ( 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 ) ) ) )
3931, 32, 38syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x
) ) )  e.  ( A  \  ( F " x ) ) ) )
4039adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
4140imp 418 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( G `
 ( A  \ 
( F " x
) ) )  e.  ( A  \  ( F " x ) ) )
4227, 41eqeltrd 2357 . . . 4  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  e.  ( A  \  ( F " x ) ) )
4342ex 423 . . 3  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
4443ralrimiva 2626 . 2  |-  ( ph  ->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
455tz7.49c 6458 . 2  |-  ( ( 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 )
461, 44, 45syl2anc 642 1  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   Oncon0 4392   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  dnnumch2  27142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388
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