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Mathbox for Glauco Siliprandi |
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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 2903 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | feqresmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2, 1 | nfres 6002 | . . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶) |
4 | feqresmptf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
5 | feqresmptf.3 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
6 | 4, 5 | fssresd 6776 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
7 | 1, 3, 6 | feqmptdf 6979 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
8 | fvres 6926 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
9 | 8 | mpteq2ia 5251 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
10 | 7, 9 | eqtrdi 2791 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2888 ⊆ wss 3963 ↦ cmpt 5231 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 |
This theorem is referenced by: (None) |
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