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Theorem f1oeq2 5061
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 5031 . . 3 (A = B → (𝐹:A1-1𝐶𝐹:B1-1𝐶))
2 foeq2 5046 . . 3 (A = B → (𝐹:Aonto𝐶𝐹:Bonto𝐶))
31, 2anbi12d 442 . 2 (A = B → ((𝐹:A1-1𝐶 𝐹:Aonto𝐶) ↔ (𝐹:B1-1𝐶 𝐹:Bonto𝐶)))
4 df-f1o 4852 . 2 (𝐹:A1-1-onto𝐶 ↔ (𝐹:A1-1𝐶 𝐹:Aonto𝐶))
5 df-f1o 4852 . 2 (𝐹:B1-1-onto𝐶 ↔ (𝐹:B1-1𝐶 𝐹:Bonto𝐶))
63, 4, 53bitr4g 212 1 (A = B → (𝐹:A1-1-onto𝐶𝐹:B1-1-onto𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  1-1wf1 4842  ontowfo 4843  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1oeq23  5063  f1oeq123d  5066  f1osng  5110  isoeq4  5387  bren  6164
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