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Theorem fneq1 5088
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
fneq1 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Proof of Theorem fneq1
StepHypRef Expression
1 funeq 5021 . . 3 (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
2 dmeq 4624 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
32eqeq1d 2096 . . 3 (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴))
41, 3anbi12d 457 . 2 (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)))
5 df-fn 5005 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6 df-fn 5005 . 2 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
74, 5, 63bitr4g 221 1 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  dom cdm 4428  Fun wfun 4996   Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004  df-fn 5005
This theorem is referenced by:  fneq1d  5090  fneq1i  5094  fn0  5119  feq1  5131  foeq1  5213  f1ocnv  5250  mpteqb  5377  eufnfv  5507  tfr0dm  6069  tfrlemiex  6078  tfr1onlemsucfn  6087  tfr1onlemsucaccv  6088  tfr1onlembxssdm  6090  tfr1onlembfn  6091  tfr1onlemex  6094  tfr1onlemaccex  6095  tfr1onlemres  6096  mapval2  6415
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