Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfmptss | Structured version Visualization version GIF version |
Description: A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
cncfmptss.1 | ⊢ Ⅎ𝑥𝐹 |
cncfmptss.2 | ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) |
cncfmptss.3 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
Ref | Expression |
---|---|
cncfmptss | ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmptss.3 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
2 | 1 | resmptd 5910 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
3 | cncfmptss.2 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) | |
4 | cncff 23503 | . . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
6 | 5 | feqmptd 6735 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) |
7 | 6 | reseq1d 5854 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶)) |
8 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
9 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑦𝑥 | |
10 | 8, 9 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
11 | cncfmptss.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
15 | 10, 13, 14 | cbvmpt 5169 | . . . 4 ⊢ (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
17 | 2, 7, 16 | 3eqtr4rd 2869 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶)) |
18 | rescncf 23507 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))) | |
19 | 1, 3, 18 | sylc 65 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)) |
20 | 17, 19 | eqeltrd 2915 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ⊆ wss 3938 ↦ cmpt 5148 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 –cn→ccncf 23486 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-cncf 23488 |
This theorem is referenced by: cncfmptssg 42160 itgsin0pilem1 42242 ibliccsinexp 42243 itgsinexplem1 42246 itgsinexp 42247 |
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