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Theorem f1oeq23 6043
 Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
Assertion
Ref Expression
f1oeq23 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))

Proof of Theorem f1oeq23
StepHypRef Expression
1 f1oeq2 6041 . 2 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
2 f1oeq3 6042 . 2 (𝐶 = 𝐷 → (𝐹:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
31, 2sylan9bb 732 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  enfixsn  7954  ackbij2lem2  8945  seqf1o  12704  eulerthlem2  15325  isgim  17527  symgval  17622  islmim  18883  fpwrelmapffs  28897  poimirlem3  32582  poimirlem15  32594  eldioph2lem1  36341
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