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Theorem fvmptf 5699
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5683 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
fvmptf.1  |-  F/_ x A
fvmptf.2  |-  F/_ x C
fvmptf.3  |-  ( x  =  A  ->  B  =  C )
fvmptf.4  |-  F  =  ( x  e.  D  |->  B )
Assertion
Ref Expression
fvmptf  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Distinct variable group:    x, D
Allowed substitution hints:    A( x)    B( x)    C( x)    F( x)    V( x)

Proof of Theorem fvmptf
StepHypRef Expression
1 elex 2872 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
2 fvmptf.1 . . . 4  |-  F/_ x A
3 fvmptf.2 . . . . . 6  |-  F/_ x C
43nfel1 2504 . . . . 5  |-  F/ x  C  e.  _V
5 fvmptf.4 . . . . . . . 8  |-  F  =  ( x  e.  D  |->  B )
6 nfmpt1 4190 . . . . . . . 8  |-  F/_ x
( x  e.  D  |->  B )
75, 6nfcxfr 2491 . . . . . . 7  |-  F/_ x F
87, 2nffv 5615 . . . . . 6  |-  F/_ x
( F `  A
)
98, 3nfeq 2501 . . . . 5  |-  F/ x
( F `  A
)  =  C
104, 9nfim 1815 . . . 4  |-  F/ x
( C  e.  _V  ->  ( F `  A
)  =  C )
11 fvmptf.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
1211eleq1d 2424 . . . . 5  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
13 fveq2 5608 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1413, 11eqeq12d 2372 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
)  =  B  <->  ( F `  A )  =  C ) )
1512, 14imbi12d 311 . . . 4  |-  ( x  =  A  ->  (
( B  e.  _V  ->  ( F `  x
)  =  B )  <-> 
( C  e.  _V  ->  ( F `  A
)  =  C ) ) )
165fvmpt2 5691 . . . . 5  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
1716ex 423 . . . 4  |-  ( x  e.  D  ->  ( B  e.  _V  ->  ( F `  x )  =  B ) )
182, 10, 15, 17vtoclgaf 2924 . . 3  |-  ( A  e.  D  ->  ( C  e.  _V  ->  ( F `  A )  =  C ) )
191, 18syl5 28 . 2  |-  ( A  e.  D  ->  ( C  e.  V  ->  ( F `  A )  =  C ) )
2019imp 418 1  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   F/_wnfc 2481   _Vcvv 2864    e. cmpt 4158   ` cfv 5337
This theorem is referenced by:  fvmptnf  5700  rdgsucmptf  6528  frsucmpt  6537  dvfsumabs  19474  dvfsumlem1  19477  dvfsumlem4  19480  dvfsum2  19485  dchrisumlem2  20751  dchrisumlem3  20752  fprodntriv  24569  prodss  24574  fprodefsum  24599  itgaddnclem2  25499  refsum2cnlem1  27031  mulc1cncfg  27044  expcnfg  27049  stoweidlem23  27095  stoweidlem26  27098  stoweidlem30  27102  stoweidlem34  27106  stoweidlem36  27108  wallispilem5  27141  stirlinglem4  27149  stirlinglem5  27150  stirlinglem8  27153  stirlinglem11  27156  stirlinglem12  27157  stirlinglem13  27158  stirlinglem14  27159  hlhilset  32196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fv 5345
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