MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fneq12 Structured version   Visualization version   GIF version

Theorem fneq12 6423
Description: Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
fneq12 ((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Proof of Theorem fneq12
StepHypRef Expression
1 simpl 486 . 2 ((𝐹 = 𝐺𝐴 = 𝐵) → 𝐹 = 𝐺)
2 simpr 488 . 2 ((𝐹 = 𝐺𝐴 = 𝐵) → 𝐴 = 𝐵)
31, 2fneq12d 6422 1 ((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538   Fn wfn 6323
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-fun 6330  df-fn 6331
This theorem is referenced by:  tfrlem3a  8000  hashresfn  13700  fprlem1  33251  frrlem15  33256
  Copyright terms: Public domain W3C validator