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Theorem fvmpt2d 5755
Description: Deduction version of fvmpt2 5753. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Hypotheses
Ref Expression
fvmpt2d.1  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
fvmpt2d.4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
Assertion
Ref Expression
fvmpt2d  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    F( x)    V( x)

Proof of Theorem fvmpt2d
StepHypRef Expression
1 fvmpt2d.1 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
21fveq1d 5672 . . 3  |-  ( ph  ->  ( F `  x
)  =  ( ( x  e.  A  |->  B ) `  x ) )
32adantr 452 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( ( x  e.  A  |->  B ) `
 x ) )
4 simpr 448 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
5 fvmpt2d.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
6 eqid 2389 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fvmpt2 5753 . . 3  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
84, 5, 7syl2anc 643 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
93, 8eqtrd 2421 1  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4209   ` cfv 5396
This theorem is referenced by:  neiptopreu  17122  ofoprabco  23923  esumcvg  24274  ofcfval2  24285  dstrvprob  24510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fv 5404
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