Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fneq1d GIF version

Theorem fneq1d 5217
 Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
fneq1d.1 (𝜑𝐹 = 𝐺)
Assertion
Ref Expression
fneq1d (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Proof of Theorem fneq1d
StepHypRef Expression
1 fneq1d.1 . 2 (𝜑𝐹 = 𝐺)
2 fneq1 5215 . 2 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
31, 2syl 14 1 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   Fn wfn 5122 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-fun 5129  df-fn 5130 This theorem is referenced by:  fneq12d  5219  f1o00  5406  f1ompt  5575  fmpt2d  5586  f1ocnvd  5976  offval2  6001  ofrfval2  6002  caofinvl  6008  f1od2  6136  cc3  7096  neif  12340
 Copyright terms: Public domain W3C validator