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Theorem dnnumch1 29350
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8192 (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 6852 . . . . . . 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 5687 . . . . . . 7  |-  ( F `
 x )  =  (recs ( ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) ) ) `  x )
53tfr1 6848 . . . . . . . . . . 11  |-  F  Fn  On
6 fnfun 5503 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Fun  F )
75, 6ax-mp 5 . . . . . . . . . 10  |-  Fun  F
8 vex 2970 . . . . . . . . . 10  |-  x  e. 
_V
9 resfunexg 5938 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
107, 8, 9mp2an 672 . . . . . . . . 9  |-  ( F  |`  x )  e.  _V
11 rneq 5060 . . . . . . . . . . . . 13  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ran  ( F  |`  x ) )
12 df-ima 4848 . . . . . . . . . . . . 13  |-  ( F
" x )  =  ran  ( F  |`  x )
1311, 12syl6eqr 2488 . . . . . . . . . . . 12  |-  ( w  =  ( F  |`  x )  ->  ran  w  =  ( F " x ) )
1413difeq2d 3469 . . . . . . . . . . 11  |-  ( w  =  ( F  |`  x )  ->  ( A  \  ran  w )  =  ( A  \ 
( F " x
) ) )
1514fveq2d 5690 . . . . . . . . . 10  |-  ( w  =  ( F  |`  x )  ->  ( G `  ( A  \  ran  w ) )  =  ( G `  ( A  \  ( F " x ) ) ) )
16 rneq 5060 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  ran  z  =  ran  w )
1716difeq2d 3469 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( A  \  ran  z )  =  ( A  \  ran  w ) )
1817fveq2d 5690 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( G `  ( A  \  ran  z ) )  =  ( G `  ( A  \  ran  w
) ) )
1918cbvmptv 4378 . . . . . . . . . 10  |-  ( z  e.  _V  |->  ( G `
 ( A  \  ran  z ) ) )  =  ( w  e. 
_V  |->  ( G `  ( A  \  ran  w
) ) )
20 fvex 5696 . . . . . . . . . 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 5 . . . . . . . 8  |-  ( ( z  e.  _V  |->  ( G `  ( A 
\  ran  z )
) ) `  ( F  |`  x ) )  =  ( G `  ( A  \  ( F " x ) ) )
233reseq1i 5101 . . . . . . . . 9  |-  ( F  |`  x )  =  (recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )  |`  x )
2423fveq2i 5689 . . . . . . . 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 726 . . . . 5  |-  ( ( ( ph  /\  x  e.  On )  /\  ( A  \  ( F "
x ) )  =/=  (/) )  ->  ( F `
 x )  =  ( G `  ( A  \  ( F "
x ) ) ) )
28 difss 3478 . . . . . . . . 9  |-  ( A 
\  ( F "
x ) )  C_  A
29 elpw2g 4450 . . . . . . . . . 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 233 . . . . . . . 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 5686 . . . . . . . . . . 11  |-  ( y  =  ( A  \ 
( F " x
) )  ->  ( G `  y )  =  ( G `  ( A  \  ( F " x ) ) ) )
35 id 22 . . . . . . . . . . 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 320 . . . . . . . . 9  |-  ( y  =  ( A  \ 
( F " x
) )  ->  (
( y  =/=  (/)  ->  ( G `  y )  e.  y )  <->  ( ( A  \  ( F "
x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) ) )
3837rspcva 3066 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x
) ) )  e.  ( A  \  ( F " x ) ) ) )
4039adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( G `  ( A  \  ( F " x ) ) )  e.  ( A 
\  ( F "
x ) ) ) )
4140imp 429 . . . . 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 434 . . 3  |-  ( (
ph  /\  x  e.  On )  ->  ( ( A  \  ( F
" x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
4443ralrimiva 2794 . 2  |-  ( ph  ->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
455tz7.49c 6893 . 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 661 1  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   _Vcvv 2967    \ cdif 3320    C_ wss 3323   (/)c0 3632   ~Pcpw 3855    e. cmpt 4345   Oncon0 4714   ran crn 4836    |` cres 4837   "cima 4838   Fun wfun 5407    Fn wfn 5408   -1-1-onto->wf1o 5412   ` cfv 5413  recscrecs 6823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6824
This theorem is referenced by:  dnnumch2  29351
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