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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 5009 | . . . 4 ⊢ ((𝐹:A⟶B ∧ 𝐶 ⊆ A) → (𝐹 ↾ 𝐶):𝐶⟶B) | |
4 | 1, 2, 3 | syl2anc 391 | . . 3 ⊢ (φ → (𝐹 ↾ 𝐶):𝐶⟶B) |
5 | 4 | feqmptd 5169 | . 2 ⊢ (φ → (𝐹 ↾ 𝐶) = (x ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘x))) |
6 | fvres 5141 | . . 3 ⊢ (x ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘x) = (𝐹‘x)) | |
7 | 6 | mpteq2ia 3834 | . 2 ⊢ (x ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘x)) = (x ∈ 𝐶 ↦ (𝐹‘x)) |
8 | 5, 7 | syl6eq 2085 | 1 ⊢ (φ → (𝐹 ↾ 𝐶) = (x ∈ 𝐶 ↦ (𝐹‘x))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ⊆ wss 2911 ↦ cmpt 3809 ↾ cres 4290 ⟶wf 4841 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 |
This theorem is referenced by: (None) |
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