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Mirrors > Home > ILE Home > Th. List > feq2 | GIF version |
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
feq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq2 4988 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | |
2 | 1 | anbi1d 438 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))) |
3 | df-f 4906 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
4 | df-f 4906 | . 2 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 212 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ⊆ wss 2917 ran crn 4346 Fn wfn 4897 ⟶wf 4898 |
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 1336 ax-gen 1338 ax-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-fn 4905 df-f 4906 |
This theorem is referenced by: feq23 5033 feq2d 5035 feq2i 5040 f00 5081 f1eq2 5088 fressnfv 5350 ac6sfi 6352 |
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