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Theorem feq2 4974
 Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq2 (A = B → (𝐹:A𝐶𝐹:B𝐶))

Proof of Theorem feq2
StepHypRef Expression
1 fneq2 4931 . . 3 (A = B → (𝐹 Fn A𝐹 Fn B))
21anbi1d 438 . 2 (A = B → ((𝐹 Fn A ran 𝐹𝐶) ↔ (𝐹 Fn B ran 𝐹𝐶)))
3 df-f 4849 . 2 (𝐹:A𝐶 ↔ (𝐹 Fn A ran 𝐹𝐶))
4 df-f 4849 . 2 (𝐹:B𝐶 ↔ (𝐹 Fn B ran 𝐹𝐶))
52, 3, 43bitr4g 212 1 (A = B → (𝐹:A𝐶𝐹:B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ⊆ wss 2911  ran crn 4289   Fn wfn 4840  ⟶wf 4841 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 This theorem is referenced by:  feq23  4976  feq2d  4978  feq2i  4983  f00  5024  f1eq2  5031  fressnfv  5293
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