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Mirrors > Home > MPE Home > Th. List > fint | Structured version Visualization version GIF version |
Description: Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fint.1 | ⊢ 𝐵 ≠ ∅ |
Ref | Expression |
---|---|
fint | ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4959 | . . . 4 ⊢ (ran 𝐹 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥) | |
2 | 1 | anbi2i 622 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥)) |
3 | fint.1 | . . . 4 ⊢ 𝐵 ≠ ∅ | |
4 | r19.28zv 4493 | . . . 4 ⊢ (𝐵 ≠ ∅ → (∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥)) |
6 | 2, 5 | bitr4i 278 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) |
7 | df-f 6538 | . 2 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵)) | |
8 | df-f 6538 | . . 3 ⊢ (𝐹:𝐴⟶𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) | |
9 | 8 | ralbii 3085 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥 ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) |
10 | 6, 7, 9 | 3bitr4i 303 | 1 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ≠ wne 2932 ∀wral 3053 ⊆ wss 3941 ∅c0 4315 ∩ cint 4941 ran crn 5668 Fn wfn 6529 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 df-int 4942 df-f 6538 |
This theorem is referenced by: chintcli 31056 |
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