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| Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6749. (Contributed by BJ, 21-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| ffrnb | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-f 6565 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 2 | dffn3 6748 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) | |
| 3 | 2 | anbi1i 624 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | 
| 4 | 1, 3 | bitri 275 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊆ wss 3951 ran crn 5686 Fn wfn 6556 ⟶wf 6557 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ss 3968 df-f 6565 | 
| This theorem is referenced by: ffrnbd 6751 | 
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