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Theorem f1oeq3d 6047
 Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
f1oeq3d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
f1oeq3d (𝜑 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))

Proof of Theorem f1oeq3d
StepHypRef Expression
1 f1oeq3d.1 . 2 (𝜑𝐴 = 𝐵)
2 f1oeq3 6042 . 2 (𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
31, 2syl 17 1 (𝜑 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = 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-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  resdif  6070  ackbij2lem2  8945  coe1mul2lem2  19459  iseupa  26492  poimirlem9  32588  sge0f1o  39275  nnfoctbdj  39349  1wlkiswwlks2lem4  41069  1wlkiswwlks2lem5  41070  eupthres  41383  eupthp1  41384
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