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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 4883 | . . . 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 4442 | . . . 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 279 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) |
7 | df-f 6352 | . 2 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵)) | |
8 | df-f 6352 | . . 3 ⊢ (𝐹:𝐴⟶𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) | |
9 | 8 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥 ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥)) |
10 | 6, 7, 9 | 3bitr4i 304 | 1 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∅c0 4288 ∩ cint 4867 ran crn 5549 Fn wfn 6343 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-int 4868 df-f 6352 |
This theorem is referenced by: chintcli 29035 |
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