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Mirrors > Home > ILE Home > Th. List > feqresmpt | GIF version |
Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
feqmptd.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
feqresmpt.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
Ref | Expression |
---|---|
feqresmpt | ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feqmptd.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | feqresmpt.2 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
3 | fssres 5086 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | |
4 | 1, 2, 3 | syl2anc 403 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
5 | 4 | feqmptd 5247 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥))) |
6 | fvres 5219 | . . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
7 | 6 | mpteq2ia 3864 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) |
8 | 5, 7 | syl6eq 2129 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ⊆ wss 2973 ↦ cmpt 3839 ↾ cres 4365 ⟶wf 4918 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 |
This theorem is referenced by: (None) |
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