New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  f1oeq23 GIF version

Theorem f1oeq23 5284
 Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
Assertion
Ref Expression
f1oeq23 ((A = B C = D) → (F:A1-1-ontoCF:B1-1-ontoD))

Proof of Theorem f1oeq23
StepHypRef Expression
1 f1oeq2 5282 . 2 (A = B → (F:A1-1-ontoCF:B1-1-ontoC))
2 f1oeq3 5283 . 2 (C = D → (F:B1-1-ontoCF:B1-1-ontoD))
31, 2sylan9bb 680 1 ((A = B C = D) → (F:A1-1-ontoCF:B1-1-ontoD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  –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-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator