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Theorem f1oeq2 5282
 Description: Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq2 (A = B → (F:A1-1-ontoCF:B1-1-ontoC))

Proof of Theorem f1oeq2
StepHypRef Expression
1 f1eq2 5254 . . 3 (A = B → (F:A1-1CF:B1-1C))
2 foeq2 5266 . . 3 (A = B → (F:AontoCF:BontoC))
31, 2anbi12d 691 . 2 (A = B → ((F:A1-1C F:AontoC) ↔ (F:B1-1C F:BontoC)))
4 df-f1o 4794 . 2 (F:A1-1-ontoC ↔ (F:A1-1C F:AontoC))
5 df-f1o 4794 . 2 (F:B1-1-ontoC ↔ (F:B1-1C F:BontoC))
63, 4, 53bitr4g 279 1 (A = B → (F:A1-1-ontoCF:B1-1-ontoC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780 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-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  f1oeq23  5284  resin  5307  f1osng  5323  isoeq4  5485  bren  6030
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