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Theorem dnnumch1 27013
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7871 (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 6625 . . . . . . 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 5692 . . . . . . 7  |-  ( F `
 x )  =  (recs ( ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) ) ) `  x )
53tfr1 6621 . . . . . . . . . . 11  |-  F  Fn  On
6 fnfun 5505 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Fun  F )
75, 6ax-mp 8 . . . . . . . . . 10  |-  Fun  F
8 vex 2923 . . . . . . . . . 10  |-  x  e. 
_V
9 resfunexg 5920 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
107, 8, 9mp2an 654 . . . . . . . . 9  |-  ( F  |`  x )  e.  _V
11 rneq 5058 . . . . . . . . . . . . 13  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ran  ( F  |`  x ) )
12 df-ima 4854 . . . . . . . . . . . . 13  |-  ( F
" x )  =  ran  ( F  |`  x )
1311, 12syl6eqr 2458 . . . . . . . . . . . 12  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ( F " x ) )
1413difeq2d 3429 . . . . . . . . . . 11  |-  ( w  =  ( F  |`  x )  ->  ( A  \  ran  w )  =  ( A  \ 
( F " x
) ) )
1514fveq2d 5695 . . . . . . . . . 10  |-  ( w  =  ( F  |`  x )  ->  ( G `  ( A  \  ran  w ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
16 rneq 5058 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  ran  z  =  ran  w )
1716difeq2d 3429 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( A  \  ran  z )  =  ( A  \  ran  w ) )
1817fveq2d 5695 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( G `  ( A  \  ran  z ) )  =  ( G `  ( A  \  ran  w
) ) )
1918cbvmptv 4264 . . . . . . . . . 10  |-  ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) )  =  ( w  e. 
_V  |->  ( G `  ( A  \  ran  w
) ) )
20 fvex 5705 . . . . . . . . . 10  |-  ( G `
 ( A  \ 
( F " x
) ) )  e. 
_V
2115, 19, 20fvmpt 5769 . . . . . . . . 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 5105 . . . . . . . . 9  |-  ( F  |`  x )  =  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x )
2423fveq2i 5694 . . . . . . . 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 2430 . . . . . . 7  |-  ( G `
 ( A  \ 
( F " x
) ) )  =  ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) `  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x ) )
262, 4, 253eqtr4g 2465 . . . . . 6  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( A  \  ( F " x ) ) ) )
2726ad2antlr 708 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  =  ( G `  ( A  \  ( F "
x ) ) ) )
28 difss 3438 . . . . . . . . 9  |-  ( A 
\  ( F "
x ) )  C_  A
29 elpw2g 4327 . . . . . . . . . 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 225 . . . . . . . 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 2579 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
y  =/=  (/)  <->  ( A  \  ( F " x
) )  =/=  (/) ) )
34 fveq2 5691 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  ( G `  y )  =  ( G `  ( A  \  ( F " x ) ) ) )
35 id 20 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  y  =  ( A  \ 
( F " x
) ) )
3634, 35eleq12d 2476 . . . . . . . . . 10  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( G `  y
)  e.  y  <->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
3733, 36imbi12d 312 . . . . . . . . 9  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  ( G `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
3837rspcva 3014 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x
) ) )  e.  ( A  \  ( F " x ) ) ) )
4039adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
4140imp 419 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( G `
 ( A  \ 
( F " x
) ) )  e.  ( A  \  ( F " x ) ) )
4227, 41eqeltrd 2482 . . . 4  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  e.  ( A  \  ( F " x ) ) )
4342ex 424 . . 3  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
4443ralrimiva 2753 . 2  |-  ( ph  ->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
455tz7.49c 6666 . 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 643 1  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671   _Vcvv 2920    \ cdif 3281    C_ wss 3284   (/)c0 3592   ~Pcpw 3763    e. cmpt 4230   Oncon0 4545   ran crn 4842    |` cres 4843   "cima 4844   Fun wfun 5411    Fn wfn 5412   -1-1-onto->wf1o 5416   ` cfv 5417  recscrecs 6595
This theorem is referenced by:  dnnumch2  27014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596
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