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Theorem fvmpt2d 6456
Description: Deduction version of fvmpt2 6454. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Hypotheses
Ref Expression
fvmpt2d.1 (𝜑𝐹 = (𝑥𝐴𝐵))
fvmpt2d.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
fvmpt2d ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpt2d
StepHypRef Expression
1 fvmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
21fveq1d 6355 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
32adantr 472 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 id 22 . . 3 (𝑥𝐴𝑥𝐴)
5 fvmpt2d.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
6 eqid 2760 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fvmpt2 6454 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
84, 5, 7syl2an2 910 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
93, 8eqtrd 2794 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cmpt 4881  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057
This theorem is referenced by:  cantnflem1  8761  frlmphl  20342  neiptopreu  21159  rrxds  23401  ofoprabco  29794  esumcvg  30478  ofcfval2  30496  eulerpartgbij  30764  dstrvprob  30863  itgexpif  31014  hgt750lemb  31064  cvgdvgrat  39032  radcnvrat  39033  binomcxplemnotnn0  39075  fmuldfeqlem1  40335  climreclmpt  40437  climinfmpt  40468  limsupubuzmpt  40472  limsupre2mpt  40483  limsupre3mpt  40487  limsupreuzmpt  40492  liminfvalxrmpt  40539  cncficcgt0  40622  dvdivbd  40659  dvnmul  40679  dvnprodlem1  40682  dvnprodlem2  40683  stoweidlem42  40780  dirkeritg  40840  elaa2lem  40971  etransclem4  40976  ioorrnopnxrlem  41047  subsaliuncllem  41096  meaiuninclem  41218  meaiininclem  41224  ovnhoilem1  41339  ovncvr2  41349  ovolval4lem1  41387  iccvonmbllem  41416  vonioolem1  41418  vonioolem2  41419  vonicclem1  41421  vonicclem2  41422  pimconstlt0  41438  pimconstlt1  41439  pimgtmnf  41456  smfpimltmpt  41479  issmfdmpt  41481  smfpimltxrmpt  41491  smfaddlem2  41496  smflimlem2  41504  smflimlem4  41506  smfpimgtmpt  41513  smfpimgtxrmpt  41516  smfmullem4  41525  smfpimcclem  41537  smfsuplem1  41541  smfsupmpt  41545  smfinfmpt  41549  smflimsuplem2  41551  smflimsuplem3  41552  smflimsuplem4  41553
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