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Theorem funeq 5127
 Description: Equality theorem for function predicate. (Contributed by set.mm contributors, 16-Aug-1994.)
Assertion
Ref Expression
funeq (A = B → (Fun A ↔ Fun B))

Proof of Theorem funeq
StepHypRef Expression
1 funss 5126 . . . 4 (B A → (Fun A → Fun B))
2 funss 5126 . . . 4 (A B → (Fun B → Fun A))
31, 2anim12i 549 . . 3 ((B A A B) → ((Fun A → Fun B) (Fun B → Fun A)))
43ancoms 439 . 2 ((A B B A) → ((Fun A → Fun B) (Fun B → Fun A)))
5 eqss 3287 . 2 (A = B ↔ (A B B A))
6 dfbi2 609 . 2 ((Fun A ↔ Fun B) ↔ ((Fun A → Fun B) (Fun B → Fun A)))
74, 5, 63imtr4i 257 1 (A = B → (Fun A ↔ Fun B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257  Fun wfun 4775 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785  df-fun 4789 This theorem is referenced by:  funeqi  5128  funeqd  5129  fununi  5160  funcnvuni  5161  cnvresid  5166  fneq1  5173  elfuns  5829  elfunsg  5830  elpmg  6013  fundmeng  6044
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