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Theorem fvmpt2d 5942
Description: Deduction version of fvmpt2 5940. (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 5850 . . 3  |-  ( ph  ->  ( F `  x
)  =  ( ( x  e.  A  |->  B ) `  x ) )
32adantr 463 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( ( x  e.  A  |->  B ) `
 x ) )
4 simpr 459 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
5 fvmpt2d.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
6 eqid 2402 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fvmpt2 5940 . . 3  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
84, 5, 7syl2anc 659 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
93, 8eqtrd 2443 1  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    |-> cmpt 4452   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fv 5576
This theorem is referenced by:  cantnflem1  8139  frlmphl  19106  neiptopreu  19925  rrxds  22115  ofoprabco  27935  esumcvg  28519  ofcfval2  28537  eulerpartgbij  28803  dstrvprob  28902  cvgdvgrat  36022  radcnvrat  36023  binomcxplemnotnn0  36089  fmuldfeqlem1  36925  cncficcgt0  37040  dvdivbd  37069  dvnmul  37089  dvnprodlem1  37092  dvnprodlem2  37093  stoweidlem42  37173  dirkeritg  37233  elaa2lem  37365  etransclem4  37370
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