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Mirrors > Home > MPE Home > Th. List > cnmpt22f | Structured version Visualization version GIF version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmpt21.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
cnmpt2t.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
cnmpt22f.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) |
Ref | Expression |
---|---|
cnmpt22f | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt21.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | cnmpt21.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
3 | cnmpt21.a | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | |
4 | cnmpt2t.b | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) | |
5 | cntop2 20855 | . . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top) | |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Top) |
7 | eqid 2610 | . . . 4 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
8 | 7 | toptopon 20548 | . . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) |
9 | 6, 8 | sylib 207 | . 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
10 | cntop2 20855 | . . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top) | |
11 | 4, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ Top) |
12 | eqid 2610 | . . . 4 ⊢ ∪ 𝑀 = ∪ 𝑀 | |
13 | 12 | toptopon 20548 | . . 3 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) |
14 | 11, 13 | sylib 207 | . 2 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) |
15 | txtopon 21204 | . . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀))) | |
16 | 9, 14, 15 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀))) |
17 | cnmpt22f.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) | |
18 | cntop2 20855 | . . . . . . . 8 ⊢ (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Top) |
20 | eqid 2610 | . . . . . . . 8 ⊢ ∪ 𝑁 = ∪ 𝑁 | |
21 | 20 | toptopon 20548 | . . . . . . 7 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) |
22 | 19, 21 | sylib 207 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
23 | cnf2 20863 | . . . . . 6 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁) | |
24 | 16, 22, 17, 23 | syl3anc 1318 | . . . . 5 ⊢ (𝜑 → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁) |
25 | ffn 5958 | . . . . 5 ⊢ (𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀)) | |
26 | 24, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀)) |
27 | fnov 6666 | . . . 4 ⊢ (𝐹 Fn (∪ 𝐿 × ∪ 𝑀) ↔ 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤))) | |
28 | 26, 27 | sylib 207 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤))) |
29 | 28, 17 | eqeltrrd 2689 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) |
30 | oveq12 6558 | . 2 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵)) | |
31 | 1, 2, 3, 4, 9, 14, 29, 30 | cnmpt22 21287 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∪ cuni 4372 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 ×t ctx 21173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-tx 21175 |
This theorem is referenced by: cnmptcom 21291 cnmpt2plusg 21702 istgp2 21705 cnmpt2vsca 21808 cnmpt2ds 22454 divcn 22479 cnrehmeo 22560 htpycom 22583 htpyco1 22585 htpycc 22587 reparphti 22605 pcohtpylem 22627 cnmpt2ip 22855 cxpcn 24286 vmcn 26938 dipcn 26959 mndpluscn 29300 cvxscon 30479 |
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