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Theorem fvmpt2d 5805
Description: Deduction version of fvmpt2 5803. (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 5721 . . 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 2435 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fvmpt2 5803 . . 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 2467 1  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5445
This theorem is referenced by:  neiptopreu  17185  ofoprabco  24067  esumcvg  24464  ofcfval2  24475  dstrvprob  24717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fv 5453
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