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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 5700 | . . . . 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 23599 | . . . . . 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 22920 | . . . . 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 4020 | . . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) |
| 13 | eqid 2737 | . . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀) | |
| 14 | 13 | cnrest 23293 | . . 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 22919 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
| 19 | 6, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) |
| 20 | topontop 22919 | . . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top) | |
| 21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top) |
| 22 | toponmax 22932 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
| 23 | 6, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
| 24 | 23, 2 | ssexd 5324 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ V) |
| 25 | toponmax 22932 | . . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀) | |
| 26 | 7, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀) |
| 27 | 26, 3 | ssexd 5324 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ V) |
| 28 | txrest 23639 | . . . . 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 2795 | . . 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 2855 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↾t crest 17465 Topctop 22899 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-map 8868 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-tx 23570 |
| This theorem is referenced by: efmndtmd 24109 submtmd 24112 iimulcn 24967 iimulcnOLD 24968 cxpcn2 26789 cxpcn3 26791 cvxsconn 35248 cvmlift2lem6 35313 cvmlift2lem12 35319 |
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