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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 6071 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶) = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
3 | cncfmptss.2 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) | |
4 | cncff 24940 | . . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
6 | 5 | feqmptd 6992 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) |
7 | 6 | reseq1d 6010 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)) ↾ 𝐶)) |
8 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
9 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑦𝑥 | |
10 | 8, 9 | nffv 6932 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
11 | cncfmptss.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6932 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | fveq2 6922 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
15 | 10, 13, 14 | cbvmpt 5277 | . . . 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 2791 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝐹 ↾ 𝐶)) |
18 | rescncf 24944 | . . 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 2844 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Ⅎwnfc 2893 ⊆ wss 3976 ↦ cmpt 5249 ↾ cres 5702 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 –cn→ccncf 24923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-map 8888 df-cncf 24925 |
This theorem is referenced by: cncfmptssg 45794 itgsin0pilem1 45873 ibliccsinexp 45874 itgsinexplem1 45877 itgsinexp 45878 |
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