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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 5750 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
3 | cncfmptss.2 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) | |
4 | cncff 23219 | . . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
6 | 5 | feqmptd 6560 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) |
7 | 6 | reseq1d 5691 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶)) |
8 | nfcv 2925 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
9 | nfcv 2925 | . . . . . 6 ⊢ Ⅎ𝑦𝑥 | |
10 | 8, 9 | nffv 6506 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
11 | cncfmptss.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2925 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6506 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | fveq2 6496 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
15 | 10, 13, 14 | cbvmpt 5023 | . . . 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 2818 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶)) |
18 | rescncf 23223 | . . 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 2859 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 Ⅎwnfc 2909 ⊆ wss 3822 ↦ cmpt 5004 ↾ cres 5405 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 –cn→ccncf 23202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-map 8206 df-cncf 23204 |
This theorem is referenced by: cncfmptssg 41617 itgsin0pilem1 41699 ibliccsinexp 41700 itgsinexplem1 41703 itgsinexp 41704 |
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