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Theorem fvmptf 5965
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5947 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 3122 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
2 fvmptf.1 . . . 4  |-  F/_ x A
3 fvmptf.2 . . . . . 6  |-  F/_ x C
43nfel1 2645 . . . . 5  |-  F/ x  C  e.  _V
5 fvmptf.4 . . . . . . . 8  |-  F  =  ( x  e.  D  |->  B )
6 nfmpt1 4536 . . . . . . . 8  |-  F/_ x
( x  e.  D  |->  B )
75, 6nfcxfr 2627 . . . . . . 7  |-  F/_ x F
87, 2nffv 5872 . . . . . 6  |-  F/_ x
( F `  A
)
98, 3nfeq 2640 . . . . 5  |-  F/ x
( F `  A
)  =  C
104, 9nfim 1867 . . . 4  |-  F/ x
( C  e.  _V  ->  ( F `  A
)  =  C )
11 fvmptf.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
1211eleq1d 2536 . . . . 5  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
13 fveq2 5865 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1413, 11eqeq12d 2489 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
)  =  B  <->  ( F `  A )  =  C ) )
1512, 14imbi12d 320 . . . 4  |-  ( x  =  A  ->  (
( B  e.  _V  ->  ( F `  x
)  =  B )  <-> 
( C  e.  _V  ->  ( F `  A
)  =  C ) ) )
165fvmpt2 5956 . . . . 5  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
1716ex 434 . . . 4  |-  ( x  e.  D  ->  ( B  e.  _V  ->  ( F `  x )  =  B ) )
182, 10, 15, 17vtoclgaf 3176 . . 3  |-  ( A  e.  D  ->  ( C  e.  _V  ->  ( F `  A )  =  C ) )
191, 18syl5 32 . 2  |-  ( A  e.  D  ->  ( C  e.  V  ->  ( F `  A )  =  C ) )
2019imp 429 1  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   _Vcvv 3113    |-> cmpt 4505   ` cfv 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fv 5595
This theorem is referenced by:  fvmptnf  5966  elfvmptrab1  5969  elovmpt3rab1  6519  rdgsucmptf  7094  frsucmpt  7103  dvfsumabs  22175  dvfsumlem1  22178  dvfsumlem4  22181  dvfsum2  22186  dchrisumlem2  23419  dchrisumlem3  23420  fprodntriv  28667  prodss  28672  fprodefsum  28697  ptrest  29641  mulc1cncfg  31155  expcnfg  31158  stoweidlem23  31339  stoweidlem34  31350  stoweidlem36  31352  wallispilem5  31385  stirlinglem4  31393  stirlinglem11  31400  stirlinglem12  31401  stirlinglem13  31402  stirlinglem14  31403  fourierdlem2  31425  fourierdlem3  31426  hlhilset  36743
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