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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 20855 | . . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Top) |
5 | eqid 2610 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
6 | 5 | toptopon 20548 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
7 | 4, 6 | sylib 207 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
8 | cnmpt11f.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) | |
9 | eqid 2610 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | 5, 9 | cnf 20860 | . . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿) |
12 | 11 | feqmptd 6159 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦))) |
13 | 12, 8 | eqeltrrd 2689 | . 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿)) |
14 | fveq2 6103 | . 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
15 | 1, 2, 7, 13, 14 | cnmpt11 21276 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∪ cuni 4372 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 |
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-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-map 7746 df-top 20521 df-topon 20523 df-cn 20841 |
This theorem is referenced by: cnmpt12f 21279 tgpmulg 21707 prdstgpd 21738 pcorevcl 22633 pcorevlem 22634 logcn 24193 loglesqrt 24299 efrlim 24496 cvmliftlem8 30528 knoppcnlem10 31662 areacirclem2 32671 areacirclem4 32673 |
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