![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmpt11f | 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 |
---|---|
cnmptid.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt11.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) |
cnmpt11f.f | ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) |
Ref | Expression |
---|---|
cnmpt11f | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmptid.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | cnmpt11.a | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) | |
3 | cntop2 23189 | . . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Top) |
5 | toptopon2 22864 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
6 | 4, 5 | sylib 217 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
7 | cnmpt11f.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) | |
8 | eqid 2725 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
9 | eqid 2725 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | 8, 9 | cnf 23194 | . . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿) |
11 | 7, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿) |
12 | 11 | feqmptd 6966 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦))) |
13 | 12, 7 | eqeltrrd 2826 | . 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿)) |
14 | fveq2 6896 | . 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
15 | 1, 2, 6, 13, 14 | cnmpt11 23611 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
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 Topctop 22839 TopOnctopon 22856 Cn ccn 23172 |
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-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-map 8847 df-top 22840 df-topon 22857 df-cn 23175 |
This theorem is referenced by: cnmpt12f 23614 tgpmulg 24041 prdstgpd 24073 pcorevcl 24996 pcorevlem 24997 logcn 26626 loglesqrt 26738 efrlim 26946 efrlimOLD 26947 cvmliftlem8 35033 knoppcnlem10 36108 areacirclem2 37313 areacirclem4 37315 |
Copyright terms: Public domain | W3C validator |