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Theorem fneq12d 6447
Description: Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
Hypotheses
Ref Expression
fneq12d.1 (𝜑𝐹 = 𝐺)
fneq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
fneq12d (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Proof of Theorem fneq12d
StepHypRef Expression
1 fneq12d.1 . . 3 (𝜑𝐹 = 𝐺)
21fneq1d 6445 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
3 fneq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43fneq2d 6446 . 2 (𝜑 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
52, 4bitrd 281 1 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533   Fn wfn 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-fun 6356  df-fn 6357
This theorem is referenced by:  fneq12  6448  seqfn  13380  sscres  17092  reschomf  17100  funcres  17165  psrvscafval  20169  ressprdsds  22980  rrxmfval  24008  ex-fpar  28240  sseqfn  31648  funcoressn  43278
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