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Theorem fneq12 4935
 Description: Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
fneq12 ((𝐹 = 𝐺 A = B) → (𝐹 Fn A𝐺 Fn B))

Proof of Theorem fneq12
StepHypRef Expression
1 simpl 102 . 2 ((𝐹 = 𝐺 A = B) → 𝐹 = 𝐺)
2 simpr 103 . 2 ((𝐹 = 𝐺 A = B) → A = B)
31, 2fneq12d 4934 1 ((𝐹 = 𝐺 A = B) → (𝐹 Fn A𝐺 Fn B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   Fn wfn 4840 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848 This theorem is referenced by:  tfrlem3ag  5865  tfrlem3a  5866  frecfnom  5925
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