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Mirrors > Home > MPE Home > Th. List > fin | Structured version Visualization version GIF version |
Description: Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fin | ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssin 4226 | . . . 4 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ↔ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)) | |
2 | 1 | anbi2i 623 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))) |
3 | anandi 674 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) | |
4 | 2, 3 | bitr3i 276 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) |
5 | df-f 6536 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝐵 ∩ 𝐶))) | |
6 | df-f 6536 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
7 | df-f 6536 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
8 | 6, 7 | anbi12i 627 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) |
9 | 4, 5, 8 | 3bitr4i 302 | 1 ⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∩ cin 3943 ⊆ wss 3944 ran crn 5670 Fn wfn 6527 ⟶wf 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-f 6536 |
This theorem is referenced by: umgrislfupgr 28248 usgrislfuspgr 28309 maprnin 31827 reprinrn 33461 reprinfz1 33465 inmap 43679 |
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