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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 2908 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | feqresmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2, 1 | nfres 5944 | . . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶) |
4 | feqresmptf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
5 | feqresmptf.3 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
6 | 4, 5 | fssresd 6714 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
7 | 1, 3, 6 | feqmptdf 6917 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
8 | fvres 6866 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
9 | 8 | mpteq2ia 5213 | . 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 1542 Ⅎwnfc 2888 ⊆ wss 3915 ↦ cmpt 5193 ↾ cres 5640 ⟶wf 6497 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 |
This theorem is referenced by: (None) |
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