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Theorem dnnumch1 27133
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7916 (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 6665 . . . . . . 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 5732 . . . . . . 7  |-  ( F `
 x )  =  (recs ( ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) ) ) `  x )
53tfr1 6661 . . . . . . . . . . 11  |-  F  Fn  On
6 fnfun 5545 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Fun  F )
75, 6ax-mp 5 . . . . . . . . . 10  |-  Fun  F
8 vex 2961 . . . . . . . . . 10  |-  x  e. 
_V
9 resfunexg 5960 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
107, 8, 9mp2an 655 . . . . . . . . 9  |-  ( F  |`  x )  e.  _V
11 rneq 5098 . . . . . . . . . . . . 13  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ran  ( F  |`  x ) )
12 df-ima 4894 . . . . . . . . . . . . 13  |-  ( F
" x )  =  ran  ( F  |`  x )
1311, 12syl6eqr 2488 . . . . . . . . . . . 12  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ( F " x ) )
1413difeq2d 3467 . . . . . . . . . . 11  |-  ( w  =  ( F  |`  x )  ->  ( A  \  ran  w )  =  ( A  \ 
( F " x
) ) )
1514fveq2d 5735 . . . . . . . . . 10  |-  ( w  =  ( F  |`  x )  ->  ( G `  ( A  \  ran  w ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
16 rneq 5098 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  ran  z  =  ran  w )
1716difeq2d 3467 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( A  \  ran  z )  =  ( A  \  ran  w ) )
1817fveq2d 5735 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( G `  ( A  \  ran  z ) )  =  ( G `  ( A  \  ran  w
) ) )
1918cbvmptv 4303 . . . . . . . . . 10  |-  ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) )  =  ( w  e. 
_V  |->  ( G `  ( A  \  ran  w
) ) )
20 fvex 5745 . . . . . . . . . 10  |-  ( G `
 ( A  \ 
( F " x
) ) )  e. 
_V
2115, 19, 20fvmpt 5809 . . . . . . . . 9  |-  ( ( F  |`  x )  e.  _V  ->  ( (
z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) `  ( F  |`  x ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
2210, 21ax-mp 5 . . . . . . . 8  |-  ( ( z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) `  ( F  |`  x ) )  =  ( G `  ( A  \  ( F " x ) ) )
233reseq1i 5145 . . . . . . . . 9  |-  ( F  |`  x )  =  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x )
2423fveq2i 5734 . . . . . . . 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 2460 . . . . . . 7  |-  ( G `
 ( A  \ 
( F " x
) ) )  =  ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) `  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x ) )
262, 4, 253eqtr4g 2495 . . . . . 6  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( A  \  ( F " x ) ) ) )
2726ad2antlr 709 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  =  ( G `  ( A  \  ( F "
x ) ) ) )
28 difss 3476 . . . . . . . . 9  |-  ( A 
\  ( F "
x ) )  C_  A
29 elpw2g 4366 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( A  \  ( F " x ) )  e.  ~P A  <->  ( A  \  ( F " x
) )  C_  A
) )
301, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  e.  ~P A 
<->  ( A  \  ( F " x ) ) 
C_  A ) )
3128, 30mpbiri 226 . . . . . . . 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 2611 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
34 fveq2 5731 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  ( G `  y )  =  ( G `  ( A  \  ( F " x ) ) ) )
35 id 21 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
3634, 35eleq12d 2506 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( G `  y
)  e.  y  <->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3733, 36imbi12d 313 . . . . . . . . 9  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  ( G `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
3837rspcva 3052 . . . . . . . 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 644 . . . . . . 7  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x
) ) )  e.  ( A  \  ( F " x ) ) ) )
4039adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
4140imp 420 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( G `
 ( A  \ 
( F " x
) ) )  e.  ( A  \  ( F " x ) ) )
4227, 41eqeltrd 2512 . . . 4  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  e.  ( A  \  ( F " x ) ) )
4342ex 425 . . 3  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
4443ralrimiva 2791 . 2  |-  ( ph  ->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
455tz7.49c 6706 . 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 644 1  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   ~Pcpw 3801    e. cmpt 4269   Oncon0 4584   ran crn 4882    |` cres 4883   "cima 4884   Fun wfun 5451    Fn wfn 5452   -1-1-onto->wf1o 5456   ` cfv 5457  recscrecs 6635
This theorem is referenced by:  dnnumch2  27134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636
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