MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpt3 Structured version   Visualization version   GIF version

Theorem fvmpt3 6428
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
fvmpt3.a (𝑥 = 𝐴𝐵 = 𝐶)
fvmpt3.b 𝐹 = (𝑥𝐷𝐵)
fvmpt3.c (𝑥𝐷𝐵𝑉)
Assertion
Ref Expression
fvmpt3 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑉
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt3
StepHypRef Expression
1 fvmpt3.a . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
21eleq1d 2835 . . 3 (𝑥 = 𝐴 → (𝐵𝑉𝐶𝑉))
3 fvmpt3.c . . 3 (𝑥𝐷𝐵𝑉)
42, 3vtoclga 3423 . 2 (𝐴𝐷𝐶𝑉)
5 fvmpt3.b . . 3 𝐹 = (𝑥𝐷𝐵)
61, 5fvmptg 6422 . 2 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
74, 6mpdan 667 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cmpt 4863  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  fvmpt3i  6429  harval  8623  mrcfval  16476  elmptrab  21851  wallispi  40804
  Copyright terms: Public domain W3C validator