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Theorem fvmptf 5966
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5946 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 3054 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
2 fvmptf.1 . . . 4  |-  F/_ x A
3 fvmptf.2 . . . . . 6  |-  F/_ x C
43nfel1 2606 . . . . 5  |-  F/ x  C  e.  _V
5 fvmptf.4 . . . . . . . 8  |-  F  =  ( x  e.  D  |->  B )
6 nfmpt1 4492 . . . . . . . 8  |-  F/_ x
( x  e.  D  |->  B )
75, 6nfcxfr 2590 . . . . . . 7  |-  F/_ x F
87, 2nffv 5872 . . . . . 6  |-  F/_ x
( F `  A
)
98, 3nfeq 2603 . . . . 5  |-  F/ x
( F `  A
)  =  C
104, 9nfim 2003 . . . 4  |-  F/ x
( C  e.  _V  ->  ( F `  A
)  =  C )
11 fvmptf.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
1211eleq1d 2513 . . . . 5  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
13 fveq2 5865 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1413, 11eqeq12d 2466 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
)  =  B  <->  ( F `  A )  =  C ) )
1512, 14imbi12d 322 . . . 4  |-  ( x  =  A  ->  (
( B  e.  _V  ->  ( F `  x
)  =  B )  <-> 
( C  e.  _V  ->  ( F `  A
)  =  C ) ) )
165fvmpt2 5957 . . . . 5  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
1716ex 436 . . . 4  |-  ( x  e.  D  ->  ( B  e.  _V  ->  ( F `  x )  =  B ) )
182, 10, 15, 17vtoclgaf 3112 . . 3  |-  ( A  e.  D  ->  ( C  e.  _V  ->  ( F `  A )  =  C ) )
191, 18syl5 33 . 2  |-  ( A  e.  D  ->  ( C  e.  V  ->  ( F `  A )  =  C ) )
2019imp 431 1  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   F/_wnfc 2579   _Vcvv 3045    |-> cmpt 4461   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590
This theorem is referenced by:  fvmptnf  5967  elfvmptrab1  5970  elovmpt3rab1  6527  rdgsucmptf  7146  frsucmpt  7155  fprodntriv  13996  prodss  14001  fprodefsum  14149  dvfsumabs  22975  dvfsumlem1  22978  dvfsumlem4  22981  dvfsum2  22986  dchrisumlem2  24328  dchrisumlem3  24329  ptrest  31939  hlhilset  35505  mulc1cncfg  37667  expcnfg  37671  stoweidlem23  37883  stoweidlem34  37895  stoweidlem36  37897  wallispilem5  37931  stirlinglem4  37939  stirlinglem11  37946  stirlinglem12  37947  stirlinglem13  37948  stirlinglem14  37949  sge0lempt  38252  sge0isummpt2  38274  meadjiun  38304
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