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Mirrors > Home > MPE Home > Th. List > ffrnbd | Structured version Visualization version GIF version |
Description: A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6528. (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 6529 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | ffrnbd.r | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) | |
3 | 2 | biantrud 535 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))) |
4 | 1, 3 | bitr4id 293 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ⊆ wss 3853 ran crn 5536 ⟶wf 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-in 3860 df-ss 3870 df-f 6353 |
This theorem is referenced by: (None) |
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