![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23188 | . . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Top) |
7 | toptopon2 22864 | . . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) | |
8 | 6, 7 | sylib 217 | . 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
9 | eqid 2725 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | eqid 2725 | . . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀 | |
11 | 9, 10 | cnf 23194 | . . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀) |
12 | 4, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀) |
13 | 12 | feqmptd 6966 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧))) |
14 | 13, 4 | eqeltrrd 2826 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀)) |
15 | fveq2 6896 | . 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴)) | |
16 | 1, 2, 3, 8, 14, 15 | cnmpt21 23619 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∪ cuni 4909 ↦ cmpt 5232 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 Topctop 22839 TopOnctopon 22856 Cn ccn 23172 ×t ctx 23508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 df-topgen 17428 df-top 22840 df-topon 22857 df-bases 22893 df-cn 23175 df-tx 23510 |
This theorem is referenced by: cnmpt22 23622 cnmptk2 23634 txhmeo 23751 tgpsubcn 24038 istgp2 24039 dvrcn 24132 htpyid 24947 htpyco1 24948 reparphti 24967 reparphtiOLD 24968 pcocn 24988 pcorevlem 24997 cxpcn 26724 cxpcnOLD 26725 dipcn 30602 mndpluscn 33658 cvxsconn 34984 cvmlift2lem6 35049 cvmlift2lem12 35055 |
Copyright terms: Public domain | W3C validator |