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Theorem fvmpt2d 6250
Description: Deduction version of fvmpt2 6248. (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 6150 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
32adantr 481 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 id 22 . . 3 (𝑥𝐴𝑥𝐴)
5 fvmpt2d.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
6 eqid 2621 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fvmpt2 6248 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
84, 5, 7syl2an2 874 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
93, 8eqtrd 2655 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cmpt 4673  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  cantnflem1  8530  frlmphl  20039  neiptopreu  20847  rrxds  23089  ofoprabco  29307  esumcvg  29929  ofcfval2  29947  eulerpartgbij  30215  dstrvprob  30314  cvgdvgrat  37994  radcnvrat  37995  binomcxplemnotnn0  38037  fmuldfeqlem1  39218  climreclmpt  39320  climinfmpt  39351  limsupubuzmpt  39355  limsupre2mpt  39366  limsupre3mpt  39370  limsupreuzmpt  39375  cncficcgt0  39405  dvdivbd  39444  dvnmul  39464  dvnprodlem1  39467  dvnprodlem2  39468  stoweidlem42  39566  dirkeritg  39626  elaa2lem  39757  etransclem4  39762  ioorrnopnxrlem  39833  subsaliuncllem  39882  meaiuninclem  40004  meaiininclem  40007  ovnhoilem1  40122  ovncvr2  40132  ovolval4lem1  40170  iccvonmbllem  40199  vonioolem1  40201  vonioolem2  40202  vonicclem1  40204  vonicclem2  40205  pimconstlt0  40221  pimconstlt1  40222  pimgtmnf  40239  smfpimltmpt  40262  issmfdmpt  40264  smfpimltxrmpt  40274  smfaddlem2  40279  smflimlem2  40287  smflimlem4  40289  smfpimgtmpt  40296  smfpimgtxrmpt  40299  smfmullem4  40308  smfpimcclem  40320  smfsuplem1  40324  smfsupmpt  40328  smfinfmpt  40332  smflimsuplem2  40334  smflimsuplem3  40335  smflimsuplem4  40336
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