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Theorem foeq2 5046
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq2 (A = B → (𝐹:Aonto𝐶𝐹:Bonto𝐶))

Proof of Theorem foeq2
StepHypRef Expression
1 fneq2 4931 . . 3 (A = B → (𝐹 Fn A𝐹 Fn B))
21anbi1d 438 . 2 (A = B → ((𝐹 Fn A ran 𝐹 = 𝐶) ↔ (𝐹 Fn B ran 𝐹 = 𝐶)))
3 df-fo 4851 . 2 (𝐹:Aonto𝐶 ↔ (𝐹 Fn A ran 𝐹 = 𝐶))
4 df-fo 4851 . 2 (𝐹:Bonto𝐶 ↔ (𝐹 Fn B ran 𝐹 = 𝐶))
52, 3, 43bitr4g 212 1 (A = B → (𝐹:Aonto𝐶𝐹:Bonto𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  ran crn 4289   Fn wfn 4840  ontowfo 4843
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-fo 4851
This theorem is referenced by:  f1oeq2  5061  foeq123d  5065  tposfo  5827
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