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Theorem feq3 4975
 Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq3 (A = B → (𝐹:𝐶A𝐹:𝐶B))

Proof of Theorem feq3
StepHypRef Expression
1 sseq2 2961 . . 3 (A = B → (ran 𝐹A ↔ ran 𝐹B))
21anbi2d 437 . 2 (A = B → ((𝐹 Fn 𝐶 ran 𝐹A) ↔ (𝐹 Fn 𝐶 ran 𝐹B)))
3 df-f 4849 . 2 (𝐹:𝐶A ↔ (𝐹 Fn 𝐶 ran 𝐹A))
4 df-f 4849 . 2 (𝐹:𝐶B ↔ (𝐹 Fn 𝐶 ran 𝐹B))
52, 3, 43bitr4g 212 1 (A = B → (𝐹:𝐶A𝐹:𝐶B))
 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-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849 This theorem is referenced by:  feq23  4976  feq123d  4980  fun2  5007  fconstg  5026  f1eq3  5032  fsng  5279  fsn2  5280  fsnunf  5305
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