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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 5572 | . . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍)) | |
5 | 2, 3, 4 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍)) |
6 | cnmpt1res.3 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
7 | cnmpt2res.8 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍)) | |
8 | txtopon 22201 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍))) | |
9 | 6, 7, 8 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍))) |
10 | toponuni 21524 | . . . . 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 4009 | . . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) |
13 | eqid 2823 | . . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀) | |
14 | 13 | cnrest 21895 | . . 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 586 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿)) |
16 | resmpo 7274 | . . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴)) | |
17 | 2, 3, 16 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴)) |
18 | topontop 21523 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 6, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | topontop 21523 | . . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top) |
22 | toponmax 21536 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
23 | 6, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
24 | 23, 2 | ssexd 5230 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ V) |
25 | toponmax 21536 | . . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀) | |
26 | 7, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀) |
27 | 26, 3 | ssexd 5230 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ V) |
28 | txrest 22241 | . . . . 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 7170 | . . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊)) |
33 | 29, 32 | syl6eqr 2876 | . . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁)) |
34 | 33 | oveq1d 7173 | . 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿)) |
35 | 15, 17, 34 | 3eltr3d 2929 | 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 2114 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ↾t crest 16696 Topctop 21503 TopOnctopon 21520 Cn ccn 21834 ×t ctx 22170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-fin 8515 df-fi 8877 df-rest 16698 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cn 21837 df-tx 22172 |
This theorem is referenced by: efmndtmd 22711 submtmd 22714 iimulcn 23544 cxpcn2 25329 cxpcn3 25331 cvxsconn 32492 cvmlift2lem6 32557 cvmlift2lem12 32563 |
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