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Theorem fvmpt2d 6556
Description: Deduction version of fvmpt2 6554. (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 6450 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
32adantr 474 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 id 22 . . 3 (𝑥𝐴𝑥𝐴)
5 fvmpt2d.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
6 eqid 2778 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fvmpt2 6554 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
84, 5, 7syl2an2 676 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
93, 8eqtrd 2814 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  cmpt 4967  cfv 6137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fv 6145
This theorem is referenced by:  cantnflem1  8885  frlmphl  20528  neiptopreu  21349  rrxds  23603  ofoprabco  30033  esumcvg  30750  ofcfval2  30768  eulerpartgbij  31036  dstrvprob  31136  itgexpif  31290  hgt750lemb  31340  cvgdvgrat  39478  radcnvrat  39479  binomcxplemnotnn0  39521  fmuldfeqlem1  40732  climreclmpt  40834  climinfmpt  40865  limsupubuzmpt  40869  limsupre2mpt  40880  limsupre3mpt  40884  limsupreuzmpt  40889  liminfvalxrmpt  40936  liminflbuz2  40965  cncficcgt0  41039  dvdivbd  41076  dvnmul  41096  dvnprodlem1  41099  dvnprodlem2  41100  stoweidlem42  41196  dirkeritg  41256  elaa2lem  41387  etransclem4  41392  ioorrnopnxrlem  41460  subsaliuncllem  41509  meaiuninclem  41631  meaiininclem  41637  ovnhoilem1  41752  ovncvr2  41762  ovolval4lem1  41800  iccvonmbllem  41829  vonioolem1  41831  vonioolem2  41832  vonicclem1  41834  vonicclem2  41835  pimconstlt0  41851  pimconstlt1  41852  pimgtmnf  41869  smfpimltmpt  41892  issmfdmpt  41894  smfpimltxrmpt  41904  smfaddlem2  41909  smflimlem2  41917  smflimlem4  41919  smfpimgtmpt  41926  smfpimgtxrmpt  41929  smfmullem4  41938  smfpimcclem  41950  smfsuplem1  41954  smfsupmpt  41958  smfinfmpt  41962  smflimsuplem2  41964  smflimsuplem3  41965  smflimsuplem4  41966
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