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Mirrors > Home > MPE Home > Th. List > cnmpt21f | 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)) |
cnmpt21f.f | ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀)) |
Ref | Expression |
---|---|
cnmpt21f | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
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 | cnmpt21f.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀)) | |
5 | cntop1 21848 | . . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Top) |
7 | toptopon2 21526 | . . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) | |
8 | 6, 7 | sylib 220 | . 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
9 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀 | |
11 | 9, 10 | cnf 21854 | . . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀) |
12 | 4, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀) |
13 | 12 | feqmptd 6733 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧))) |
14 | 13, 4 | eqeltrrd 2914 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀)) |
15 | fveq2 6670 | . 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴)) | |
16 | 1, 2, 3, 8, 14, 15 | cnmpt21 22279 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∪ cuni 4838 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Topctop 21501 TopOnctopon 21518 Cn ccn 21832 ×t ctx 22168 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cn 21835 df-tx 22170 |
This theorem is referenced by: cnmpt22 22282 cnmptk2 22294 txhmeo 22411 tgpsubcn 22698 istgp2 22699 dvrcn 22792 htpyid 23581 htpyco1 23582 reparphti 23601 pcocn 23621 pcorevlem 23630 cxpcn 25326 dipcn 28497 mndpluscn 31169 cvxsconn 32490 cvmlift2lem6 32555 cvmlift2lem12 32561 |
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