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Theorem f1oeq3 5283
 Description: Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq3 (A = B → (F:C1-1-ontoAF:C1-1-ontoB))

Proof of Theorem f1oeq3
StepHypRef Expression
1 f1eq3 5255 . . 3 (A = B → (F:C1-1AF:C1-1B))
2 foeq3 5267 . . 3 (A = B → (F:ContoAF:ContoB))
31, 2anbi12d 691 . 2 (A = B → ((F:C1-1A F:ContoA) ↔ (F:C1-1B F:ContoB)))
4 df-f1o 4794 . 2 (F:C1-1-ontoA ↔ (F:C1-1A F:ContoA))
5 df-f1o 4794 . 2 (F:C1-1-ontoB ↔ (F:C1-1B F:ContoB))
63, 4, 53bitr4g 279 1 (A = B → (F:C1-1-ontoAF:C1-1-ontoB))
 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-mp 5  ax-1 6  ax-2 7  ax-3 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-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  f1oeq23  5284  resdif  5306  resin  5307  f1osng  5323  isoeq5  5486  isoini2  5498  swapres  5512  bren  6030  xpcomen  6052  xpassen  6057  enpw1pw  6075
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