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Theorem fvmpt2d 5404
Description: Deduction version of fvmpt2 5401. (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 5322 . . 3  |-  ( ph  ->  ( F `  x
)  =  ( ( x  e.  A  |->  B ) `  x ) )
32adantr 271 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( ( x  e.  A  |->  B ) `
 x ) )
4 simpr 109 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
5 fvmpt2d.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
6 eqid 2089 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fvmpt2 5401 . . 3  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
84, 5, 7syl2anc 404 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
93, 8eqtrd 2121 1  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439    |-> cmpt 3907   ` cfv 5030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038
This theorem is referenced by:  iseqf1olemjpcl  9987  iseqf1olemqpcl  9988  isumshft  10947
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