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Theorem fvmpt2d 6774
Description: Deduction version of fvmpt2 6772. (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 6665 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
32adantr 484 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 id 22 . . 3 (𝑥𝐴𝑥𝐴)
5 fvmpt2d.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
6 eqid 2824 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fvmpt2 6772 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
84, 5, 7syl2an2 685 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
93, 8eqtrd 2859 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cmpt 5133  cfv 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fv 6353
This theorem is referenced by:  cantnflem1  9151  frlmphl  20479  neiptopreu  21747  rrxds  24006  ofoprabco  30428  suppovss  30445  tocycf  30801  fedgmullem2  31094  esumcvg  31430  ofcfval2  31448  eulerpartgbij  31715  dstrvprob  31814  itgexpif  31962  hgt750lemb  32012  frlmsnic  39414  cvgdvgrat  40965  radcnvrat  40966  binomcxplemnotnn0  41008  fmuldfeqlem1  42177  climreclmpt  42279  climinfmpt  42310  limsupubuzmpt  42314  limsupre2mpt  42325  limsupre3mpt  42329  limsupreuzmpt  42334  liminfvalxrmpt  42381  liminflbuz2  42410  cncficcgt0  42483  dvdivbd  42518  dvnmul  42538  dvnprodlem1  42541  dvnprodlem2  42542  stoweidlem42  42637  dirkeritg  42697  elaa2lem  42828  etransclem4  42833  ioorrnopnxrlem  42901  subsaliuncllem  42950  meaiuninclem  43072  meaiininclem  43078  ovnhoilem1  43193  ovncvr2  43203  ovolval4lem1  43241  iccvonmbllem  43270  vonioolem1  43272  vonioolem2  43273  vonicclem1  43275  vonicclem2  43276  pimconstlt0  43292  pimconstlt1  43293  pimgtmnf  43310  smfpimltmpt  43333  issmfdmpt  43335  smfpimltxrmpt  43345  smfaddlem2  43350  smflimlem2  43358  smflimlem4  43360  smfpimgtmpt  43367  smfpimgtxrmpt  43370  smfmullem4  43379  smfpimcclem  43391  smfsuplem1  43395  smfsupmpt  43399  smfinfmpt  43403  smflimsuplem2  43405  smflimsuplem3  43406  smflimsuplem4  43407
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