Theorem fneq1d 5349

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 5347	. 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 105   = wceq 1364   Fn wfn 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-fun 5261  df-fn 5262

This theorem is referenced by:  fneq12d  5351  f1o00  5542  f1ompt  5716  fmpt2d  5727  f1ocnvd  6129  offval2  6155  ofrfval2  6156  caofinvl  6165  f1od2  6302  cc3  7351  plusffng  13067  grpinvfng  13246  grpinvf1o  13272  mulgfng  13330  srg1zr  13619  scaffng  13941  neif  14461  fnmptd  15534