Theorem fneq1d 5272

Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
fneq1d.1	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
fneq1d	⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))

Proof of Theorem fneq1d

Step	Hyp	Ref	Expression
1		fneq1d.1	. 2 ⊢ (𝜑 → 𝐹 = 𝐺)
2		fneq1 5270	. 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1342   Fn wfn 5177

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-fun 5184  df-fn 5185

This theorem is referenced by:  fneq12d  5274  f1o00  5461  f1ompt  5630  fmpt2d  5641  f1ocnvd  6034  offval2  6059  ofrfval2  6060  caofinvl  6066  f1od2  6194  cc3  7200  neif  12682  fnmptd  13521