| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > feqresmptf | Structured version Visualization version GIF version | ||
| Description: Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| feqresmptf.1 | ⊢ Ⅎ𝑥𝐹 |
| feqresmptf.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| feqresmptf.3 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
| Ref | Expression |
|---|---|
| feqresmptf | ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
| 2 | feqresmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
| 3 | 2, 1 | nfres 5999 | . . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶) |
| 4 | feqresmptf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 5 | feqresmptf.3 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
| 6 | 4, 5 | fssresd 6775 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
| 7 | 1, 3, 6 | feqmptdf 6979 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
| 8 | fvres 6925 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
| 9 | 8 | mpteq2ia 5245 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
| 10 | 7, 9 | eqtrdi 2793 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnfc 2890 ⊆ wss 3951 ↦ cmpt 5225 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |