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Mirrors > Home > MPE Home > Th. List > cnmpt2res | Structured version Visualization version GIF version |
Description: The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.) |
Ref | Expression |
---|---|
cnmpt1res.2 | ⊢ 𝐾 = (𝐽 ↾t 𝑌) |
cnmpt1res.3 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt1res.5 | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
cnmpt2res.7 | ⊢ 𝑁 = (𝑀 ↾t 𝑊) |
cnmpt2res.8 | ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍)) |
cnmpt2res.9 | ⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
cnmpt2res.10 | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿)) |
Ref | Expression |
---|---|
cnmpt2res | ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt2res.10 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿)) | |
2 | cnmpt1res.5 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
3 | cnmpt2res.9 | . . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍) | |
4 | xpss12 5534 | . . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍)) | |
5 | 2, 3, 4 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍)) |
6 | cnmpt1res.3 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
7 | cnmpt2res.8 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍)) | |
8 | txtopon 22335 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍))) | |
9 | 6, 7, 8 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍))) |
10 | toponuni 21658 | . . . . 5 ⊢ ((𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)) → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀)) |
12 | 5, 11 | sseqtrd 3915 | . . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) |
13 | eqid 2738 | . . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀) | |
14 | 13 | cnrest 22029 | . . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿) ∧ (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿)) |
15 | 1, 12, 14 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿)) |
16 | resmpo 7280 | . . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴)) | |
17 | 2, 3, 16 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴)) |
18 | topontop 21657 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 6, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | topontop 21657 | . . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top) |
22 | toponmax 21670 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
23 | 6, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
24 | 23, 2 | ssexd 5189 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ V) |
25 | toponmax 21670 | . . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀) | |
26 | 7, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀) |
27 | 26, 3 | ssexd 5189 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ V) |
28 | txrest 22375 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑀 ∈ Top) ∧ (𝑌 ∈ V ∧ 𝑊 ∈ V)) → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))) | |
29 | 19, 21, 24, 27, 28 | syl22anc 838 | . . . 4 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))) |
30 | cnmpt1res.2 | . . . . 5 ⊢ 𝐾 = (𝐽 ↾t 𝑌) | |
31 | cnmpt2res.7 | . . . . 5 ⊢ 𝑁 = (𝑀 ↾t 𝑊) | |
32 | 30, 31 | oveq12i 7176 | . . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊)) |
33 | 29, 32 | eqtr4di 2791 | . . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁)) |
34 | 33 | oveq1d 7179 | . 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿)) |
35 | 15, 17, 34 | 3eltr3d 2847 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 Vcvv 3397 ⊆ wss 3841 ∪ cuni 4793 × cxp 5517 ↾ cres 5521 ‘cfv 6333 (class class class)co 7164 ∈ cmpo 7166 ↾t crest 16790 Topctop 21637 TopOnctopon 21654 Cn ccn 21968 ×t ctx 22304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-map 8432 df-en 8549 df-fin 8552 df-fi 8941 df-rest 16792 df-topgen 16813 df-top 21638 df-topon 21655 df-bases 21690 df-cn 21971 df-tx 22306 |
This theorem is referenced by: efmndtmd 22845 submtmd 22848 iimulcn 23683 cxpcn2 25479 cxpcn3 25481 cvxsconn 32768 cvmlift2lem6 32833 cvmlift2lem12 32839 |
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