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Theorem dnnumch1 35598
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8459 (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 7130 . . . . . . 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 5882 . . . . . . 7  |-  ( F `
 x )  =  (recs ( ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) ) ) `  x )
53tfr1 7123 . . . . . . . . . . 11  |-  F  Fn  On
6 fnfun 5691 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Fun  F )
75, 6ax-mp 5 . . . . . . . . . 10  |-  Fun  F
8 vex 3090 . . . . . . . . . 10  |-  x  e. 
_V
9 resfunexg 6145 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
107, 8, 9mp2an 676 . . . . . . . . 9  |-  ( F  |`  x )  e.  _V
11 rneq 5080 . . . . . . . . . . . . 13  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ran  ( F  |`  x ) )
12 df-ima 4867 . . . . . . . . . . . . 13  |-  ( F
" x )  =  ran  ( F  |`  x )
1311, 12syl6eqr 2488 . . . . . . . . . . . 12  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ( F " x ) )
1413difeq2d 3589 . . . . . . . . . . 11  |-  ( w  =  ( F  |`  x )  ->  ( A  \  ran  w )  =  ( A  \ 
( F " x
) ) )
1514fveq2d 5885 . . . . . . . . . 10  |-  ( w  =  ( F  |`  x )  ->  ( G `  ( A  \  ran  w ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
16 rneq 5080 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  ran  z  =  ran  w )
1716difeq2d 3589 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( A  \  ran  z )  =  ( A  \  ran  w ) )
1817fveq2d 5885 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( G `  ( A  \  ran  z ) )  =  ( G `  ( A  \  ran  w
) ) )
1918cbvmptv 4518 . . . . . . . . . 10  |-  ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) )  =  ( w  e. 
_V  |->  ( G `  ( A  \  ran  w
) ) )
20 fvex 5891 . . . . . . . . . 10  |-  ( G `
 ( A  \ 
( F " x
) ) )  e. 
_V
2115, 19, 20fvmpt 5964 . . . . . . . . 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 5121 . . . . . . . . 9  |-  ( F  |`  x )  =  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x )
2423fveq2i 5884 . . . . . . . 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 731 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  =  ( G `  ( A  \  ( F "
x ) ) ) )
28 difss 3598 . . . . . . . . 9  |-  ( A 
\  ( F "
x ) )  C_  A
29 elpw2g 4588 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( A  \  ( F " x ) )  e.  ~P A  <->  ( A  \  ( F " x
) )  C_  A
) )
301, 29syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  e.  ~P A 
<->  ( A  \  ( F " x ) ) 
C_  A ) )
3128, 30mpbiri 236 . . . . . . . 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 2712 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
34 fveq2 5881 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  ( G `  y )  =  ( G `  ( A  \  ( F " x ) ) ) )
35 id 23 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
3634, 35eleq12d 2511 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( G `  y
)  e.  y  <->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3733, 36imbi12d 321 . . . . . . . . 9  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  ( G `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
3837rspcva 3186 . . . . . . . 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 665 . . . . . . 7  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x
) ) )  e.  ( A  \  ( F " x ) ) ) )
4039adantr 466 . . . . . 6  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
4140imp 430 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( G `
 ( A  \ 
( F " x
) ) )  e.  ( A  \  ( F " x ) ) )
4227, 41eqeltrd 2517 . . . 4  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  e.  ( A  \  ( F " x ) ) )
4342ex 435 . . 3  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
4443ralrimiva 2846 . 2  |-  ( ph  ->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
455tz7.49c 7171 . 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 665 1  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   _Vcvv 3087    \ cdif 3439    C_ wss 3442   (/)c0 3767   ~Pcpw 3985    |-> cmpt 4484   ran crn 4855    |` cres 4856   "cima 4857   Oncon0 5442   Fun wfun 5595    Fn wfn 5596   -1-1-onto->wf1o 5600   ` cfv 5601  recscrecs 7097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-wrecs 7036  df-recs 7098
This theorem is referenced by:  dnnumch2  35599
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