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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 5604 | . . . . 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 22742 | . . . . . 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 22063 | . . . . 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 3961 | . . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) |
13 | eqid 2738 | . . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀) | |
14 | 13 | cnrest 22436 | . . 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 7394 | . . 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 22062 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 6, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | topontop 22062 | . . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top) |
22 | toponmax 22075 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
23 | 6, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
24 | 23, 2 | ssexd 5248 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ V) |
25 | toponmax 22075 | . . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀) | |
26 | 7, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀) |
27 | 26, 3 | ssexd 5248 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ V) |
28 | txrest 22782 | . . . . 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 836 | . . . 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 7287 | . . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊)) |
33 | 29, 32 | eqtr4di 2796 | . . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁)) |
34 | 33 | oveq1d 7290 | . 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 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∪ cuni 4839 × cxp 5587 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ↾t crest 17131 Topctop 22042 TopOnctopon 22059 Cn ccn 22375 ×t ctx 22711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-map 8617 df-en 8734 df-fin 8737 df-fi 9170 df-rest 17133 df-topgen 17154 df-top 22043 df-topon 22060 df-bases 22096 df-cn 22378 df-tx 22713 |
This theorem is referenced by: efmndtmd 23252 submtmd 23255 iimulcn 24101 cxpcn2 25899 cxpcn3 25901 cvxsconn 33205 cvmlift2lem6 33270 cvmlift2lem12 33276 |
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