ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvmpt3i GIF version

Theorem fvmpt3i 5549
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvmpt3.a (𝑥 = 𝐴𝐵 = 𝐶)
fvmpt3.b 𝐹 = (𝑥𝐷𝐵)
fvmpt3i.c 𝐵 ∈ V
Assertion
Ref Expression
fvmpt3i (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt3i
StepHypRef Expression
1 fvmpt3.a . 2 (𝑥 = 𝐴𝐵 = 𝐶)
2 fvmpt3.b . 2 𝐹 = (𝑥𝐷𝐵)
3 fvmpt3i.c . . 3 𝐵 ∈ V
43a1i 9 . 2 (𝑥𝐷𝐵 ∈ V)
51, 2, 4fvmpt3 5548 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  Vcvv 2712  cmpt 4026  cfv 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179
This theorem is referenced by:  flval  10175  0tonninf  10342  1tonninf  10343  istopon  12453
  Copyright terms: Public domain W3C validator