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Mirrors > Home > MPE Home > Th. List > ffrnb | Structured version Visualization version GIF version |
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6548. (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 6373 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | dffn3 6547 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) | |
3 | 2 | anbi1i 627 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
4 | 1, 3 | bitri 278 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ⊆ wss 3857 ran crn 5541 Fn wfn 6364 ⟶wf 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-v 3403 df-in 3864 df-ss 3874 df-f 6373 |
This theorem is referenced by: ffrnbd 6550 |
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