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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 21415 | . . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Top) |
7 | eqid 2825 | . . . 4 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
8 | 7 | toptopon 21092 | . . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) |
9 | 6, 8 | sylib 210 | . 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
10 | eqid 2825 | . . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀 | |
11 | 7, 10 | cnf 21421 | . . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀) |
12 | 4, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀) |
13 | 12 | feqmptd 6496 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧))) |
14 | 13, 4 | eqeltrrd 2907 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀)) |
15 | fveq2 6433 | . 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴)) | |
16 | 1, 2, 3, 9, 14, 15 | cnmpt21 21845 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ∪ cuni 4658 ↦ cmpt 4952 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 Topctop 21068 TopOnctopon 21085 Cn ccn 21399 ×t ctx 21734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-map 8124 df-topgen 16457 df-top 21069 df-topon 21086 df-bases 21121 df-cn 21402 df-tx 21736 |
This theorem is referenced by: cnmpt22 21848 cnmptk2 21860 txhmeo 21977 tgpsubcn 22264 istgp2 22265 dvrcn 22357 htpyid 23146 htpyco1 23147 reparphti 23166 pcocn 23186 pcorevlem 23195 cxpcn 24888 dipcn 28130 mndpluscn 30517 cvxsconn 31771 cvmlift2lem6 31836 cvmlift2lem12 31842 |
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