MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpts Structured version   Visualization version   GIF version

Theorem fvmpts 7001
Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fvmpts.1 𝐹 = (𝑥𝐶𝐵)
Assertion
Ref Expression
fvmpts ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpts
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3896 . 2 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
2 fvmpts.1 . . 3 𝐹 = (𝑥𝐶𝐵)
3 nfcv 2903 . . . 4 𝑦𝐵
4 nfcsb1v 3918 . . . 4 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3907 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 5259 . . 3 (𝑥𝐶𝐵) = (𝑦𝐶𝑦 / 𝑥𝐵)
72, 6eqtri 2760 . 2 𝐹 = (𝑦𝐶𝑦 / 𝑥𝐵)
81, 7fvmptg 6996 1 ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  csb 3893  cmpt 5231  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  fvmptdf  7004  fvmpocurryd  8255  mptnn0fsupp  13961  mptnn0fsuppr  13963  zsum  15663  prodss  15890  fprodser  15892  fprodn0  15922  fprodefsum  16037  pcmpt  16824  issubc  17784  gsummptnn0fz  19853  mptscmfsupp0  20536  gsummoncoe1  21827  fvmptnn04if  22350  prdsdsf  23872  itgparts  25563  dchrisumlema  26988  abfmpeld  31874  abfmpel  31875  cdlemk40  39783  aomclem6  41791  ellimcabssub0  44323  constlimc  44330  vonn0ioo2  45396  vonn0icc2  45398
  Copyright terms: Public domain W3C validator