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Theorem fvmpt2d 5959
Description: Deduction version of fvmpt2 5957. (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 5867 . . 3  |-  ( ph  ->  ( F `  x
)  =  ( ( x  e.  A  |->  B ) `  x ) )
32adantr 467 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( ( x  e.  A  |->  B ) `
 x ) )
4 simpr 463 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
5 fvmpt2d.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
6 eqid 2451 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fvmpt2 5957 . . 3  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
84, 5, 7syl2anc 667 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
93, 8eqtrd 2485 1  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    |-> 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:  cantnflem1  8194  frlmphl  19339  neiptopreu  20149  rrxds  22352  ofoprabco  28267  esumcvg  28907  ofcfval2  28925  eulerpartgbij  29205  dstrvprob  29304  cvgdvgrat  36662  radcnvrat  36663  binomcxplemnotnn0  36705  fmuldfeqlem1  37660  cncficcgt0  37766  dvdivbd  37795  dvnmul  37818  dvnprodlem1  37821  dvnprodlem2  37822  stoweidlem42  37903  dirkeritg  37964  elaa2lem  38097  elaa2lemOLD  38098  etransclem4  38103  ovnhoilem1  38423  ovncvr2  38433
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