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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 2977 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | feqresmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2, 1 | nfres 5855 | . . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶) |
4 | feqresmptf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
5 | feqresmptf.3 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
6 | 4, 5 | fssresd 6545 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
7 | 1, 3, 6 | feqmptdf 6735 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
8 | fvres 6689 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
9 | 8 | mpteq2ia 5157 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
10 | 7, 9 | syl6eq 2872 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2961 ⊆ wss 3936 ↦ cmpt 5146 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 |
This theorem is referenced by: (None) |
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