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Theorem f1oeq2 5146
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq2 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))

Proof of Theorem f1oeq2
StepHypRef Expression
1 f1eq2 5116 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
2 foeq2 5131 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
31, 2anbi12d 450 . 2 (𝐴 = 𝐵 → ((𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶) ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶)))
4 df-f1o 4937 . 2 (𝐹:𝐴1-1-onto𝐶 ↔ (𝐹:𝐴1-1𝐶𝐹:𝐴onto𝐶))
5 df-f1o 4937 . 2 (𝐹:𝐵1-1-onto𝐶 ↔ (𝐹:𝐵1-1𝐶𝐹:𝐵onto𝐶))
63, 4, 53bitr4g 216 1 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  1-1wf1 4927  ontowfo 4928  1-1-ontowf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  f1oeq23  5148  f1oeq123d  5151  f1osng  5195  isoeq4  5472  bren  6259
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