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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 4962 | . . . 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 4496 | . . . 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 6546 | . 2 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵)) | |
8 | df-f 6546 | . . 3 ⊢ (𝐹:𝐴⟶𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) | |
9 | 8 | ralbii 3089 | . 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 2936 ∀wral 3057 ⊆ wss 3945 ∅c0 4318 ∩ cint 4944 ran crn 5673 Fn wfn 6537 ⟶wf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-v 3472 df-dif 3948 df-in 3952 df-ss 3962 df-nul 4319 df-int 4945 df-f 6546 |
This theorem is referenced by: chintcli 31134 |
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