Theorem fneq1d 5420

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 5418	. 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 1397   Fn wfn 5321

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-fun 5328  df-fn 5329

This theorem is referenced by:  fneq12d  5422  f1o00  5620  f1ompt  5798  fmpt2d  5809  f1ocnvd  6225  offval2  6251  ofrfval2  6252  caofinvl  6261  f1od2  6400  cc3  7487  ccatvalfn  11182  swrdlen  11237  plusffng  13453  grpinvfng  13632  grpinvf1o  13658  mulgfng  13716  srg1zr  14006  scaffng  14329  neif  14871  fnmptd  16426