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Theorem fvmptf 5588
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5572 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 2741 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
2 fvmptf.1 . . . 4  |-  F/_ x A
3 fvmptf.2 . . . . . 6  |-  F/_ x C
43nfel1 2323 . . . . 5  |-  F/ x  C  e.  _V
5 fvmptf.4 . . . . . . . 8  |-  F  =  ( x  e.  D  |->  B )
6 nfmpt1 4082 . . . . . . . 8  |-  F/_ x
( x  e.  D  |->  B )
75, 6nfcxfr 2309 . . . . . . 7  |-  F/_ x F
87, 2nffv 5506 . . . . . 6  |-  F/_ x
( F `  A
)
98, 3nfeq 2320 . . . . 5  |-  F/ x
( F `  A
)  =  C
104, 9nfim 1565 . . . 4  |-  F/ x
( C  e.  _V  ->  ( F `  A
)  =  C )
11 fvmptf.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
1211eleq1d 2239 . . . . 5  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
13 fveq2 5496 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1413, 11eqeq12d 2185 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
)  =  B  <->  ( F `  A )  =  C ) )
1512, 14imbi12d 233 . . . 4  |-  ( x  =  A  ->  (
( B  e.  _V  ->  ( F `  x
)  =  B )  <-> 
( C  e.  _V  ->  ( F `  A
)  =  C ) ) )
165fvmpt2 5579 . . . . 5  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
1716ex 114 . . . 4  |-  ( x  e.  D  ->  ( B  e.  _V  ->  ( F `  x )  =  B ) )
182, 10, 15, 17vtoclgaf 2795 . . 3  |-  ( A  e.  D  ->  ( C  e.  _V  ->  ( F `  A )  =  C ) )
191, 18syl5 32 . 2  |-  ( A  e.  D  ->  ( C  e.  V  ->  ( F `  A )  =  C ) )
2019imp 123 1  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   F/_wnfc 2299   _Vcvv 2730    |-> cmpt 4050   ` cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206
This theorem is referenced by:  fvmptd3  5589  elfvmptrab1  5590  sumrbdclem  11340  fsum3  11350  isumss  11354  prodrbdclem  11534  prodmodclem2a  11539  zproddc  11542  fprodntrivap  11547  prodssdc  11552  pcmpt  12295
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