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Theorem funeq 4864
Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
funeq (A = B → (Fun A ↔ Fun B))

Proof of Theorem funeq
StepHypRef Expression
1 eqimss2 2992 . . 3 (A = BBA)
2 funss 4863 . . 3 (BA → (Fun A → Fun B))
31, 2syl 14 . 2 (A = B → (Fun A → Fun B))
4 eqimss 2991 . . 3 (A = BAB)
5 funss 4863 . . 3 (AB → (Fun B → Fun A))
64, 5syl 14 . 2 (A = B → (Fun B → Fun A))
73, 6impbid 120 1 (A = B → (Fun A ↔ Fun B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wss 2911  Fun wfun 4839
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-bnd 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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847
This theorem is referenced by:  funeqi  4865  funeqd  4866  fununi  4910  funcnvuni  4911  cnvresid  4916  fneq1  4930  fundmeng  6223
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