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| Description: A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn 6748. (Contributed by AV, 20-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| ffrnbd.r | ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) | 
| Ref | Expression | 
|---|---|
| ffrnbd | ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ffrnb 6749 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 2 | ffrnbd.r | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) | |
| 3 | 2 | biantrud 531 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))) | 
| 4 | 1, 3 | bitr4id 290 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ⊆ wss 3950 ran crn 5685 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ss 3967 df-f 6564 | 
| This theorem is referenced by: (None) | 
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