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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 2892 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | feqresmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2, 1 | nfres 5981 | . . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶) |
4 | feqresmptf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
5 | feqresmptf.3 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
6 | 4, 5 | fssresd 6759 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
7 | 1, 3, 6 | feqmptdf 6964 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
8 | fvres 6911 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
9 | 8 | mpteq2ia 5246 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
10 | 7, 9 | eqtrdi 2781 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 2875 ⊆ wss 3939 ↦ cmpt 5226 ↾ cres 5674 ⟶wf 6539 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 |
This theorem is referenced by: (None) |
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