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Theorem fneq1d 4989
 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 4987 . 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 98   = wceq 1243   Fn wfn 4897 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-fn 4905 This theorem is referenced by:  fneq12d  4991  f1o00  5161  f1ompt  5320  fmpt2d  5327  f1ocnvd  5702  offval2  5726  ofrfval2  5727  caofinvl  5733
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