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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 5704 | . . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍)) | |
5 | 2, 3, 4 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍)) |
6 | cnmpt1res.3 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
7 | cnmpt2res.8 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍)) | |
8 | txtopon 23615 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍))) | |
9 | 6, 7, 8 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍))) |
10 | toponuni 22936 | . . . . 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 4036 | . . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) |
13 | eqid 2735 | . . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀) | |
14 | 13 | cnrest 23309 | . . 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 584 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿)) |
16 | resmpo 7553 | . . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴)) | |
17 | 2, 3, 16 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴)) |
18 | topontop 22935 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 6, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | topontop 22935 | . . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top) |
22 | toponmax 22948 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
23 | 6, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
24 | 23, 2 | ssexd 5330 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ V) |
25 | toponmax 22948 | . . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀) | |
26 | 7, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀) |
27 | 26, 3 | ssexd 5330 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ V) |
28 | txrest 23655 | . . . . 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 839 | . . . 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 7443 | . . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊)) |
33 | 29, 32 | eqtr4di 2793 | . . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁)) |
34 | 33 | oveq1d 7446 | . 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿)) |
35 | 15, 17, 34 | 3eltr3d 2853 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 × cxp 5687 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↾t crest 17467 Topctop 22915 TopOnctopon 22932 Cn ccn 23248 ×t ctx 23584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-map 8867 df-en 8985 df-fin 8988 df-fi 9449 df-rest 17469 df-topgen 17490 df-top 22916 df-topon 22933 df-bases 22969 df-cn 23251 df-tx 23586 |
This theorem is referenced by: efmndtmd 24125 submtmd 24128 iimulcn 24981 iimulcnOLD 24982 cxpcn2 26804 cxpcn3 26806 cvxsconn 35228 cvmlift2lem6 35293 cvmlift2lem12 35299 |
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