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Theorem fneq12 6638
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 482 . 2 ((𝐹 = 𝐺𝐴 = 𝐵) → 𝐹 = 𝐺)
2 simpr 484 . 2 ((𝐹 = 𝐺𝐴 = 𝐵) → 𝐴 = 𝐵)
31, 2fneq12d 6637 1 ((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533   Fn wfn 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-fun 6538  df-fn 6539
This theorem is referenced by:  fprlem1  8283  tfrlem3a  8375  frrlem15  9751  hashresfn  14303
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