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

Theorem fvmpts 7002
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 3897 . 2 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
2 fvmpts.1 . . 3 𝐹 = (𝑥𝐶𝐵)
3 nfcv 2904 . . . 4 𝑦𝐵
4 nfcsb1v 3919 . . . 4 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3908 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 5260 . . 3 (𝑥𝐶𝐵) = (𝑦𝐶𝑦 / 𝑥𝐵)
72, 6eqtri 2761 . 2 𝐹 = (𝑦𝐶𝑦 / 𝑥𝐵)
81, 7fvmptg 6997 1 ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  csb 3894  cmpt 5232  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552
This theorem is referenced by:  fvmptdf  7005  fvmpocurryd  8256  mptnn0fsupp  13962  mptnn0fsuppr  13964  zsum  15664  prodss  15891  fprodser  15893  fprodn0  15923  fprodefsum  16038  pcmpt  16825  issubc  17785  gsummptnn0fz  19854  mptscmfsupp0  20537  gsummoncoe1  21828  fvmptnn04if  22351  prdsdsf  23873  itgparts  25564  dchrisumlema  26991  abfmpeld  31879  abfmpel  31880  cdlemk40  39788  aomclem6  41801  ellimcabssub0  44333  constlimc  44340  vonn0ioo2  45406  vonn0icc2  45408
  Copyright terms: Public domain W3C validator