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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 21849 | . . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Top) |
5 | toptopon2 21526 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
6 | 4, 5 | sylib 220 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
7 | cnmpt11f.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) | |
8 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
9 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | 8, 9 | cnf 21854 | . . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿) |
11 | 7, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿) |
12 | 11 | feqmptd 6733 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦))) |
13 | 12, 7 | eqeltrrd 2914 | . 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿)) |
14 | fveq2 6670 | . 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
15 | 1, 2, 6, 13, 14 | cnmpt11 22271 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
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 Topctop 21501 TopOnctopon 21518 Cn ccn 21832 |
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-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-map 8408 df-top 21502 df-topon 21519 df-cn 21835 |
This theorem is referenced by: cnmpt12f 22274 tgpmulg 22701 prdstgpd 22733 pcorevcl 23629 pcorevlem 23630 logcn 25230 loglesqrt 25339 efrlim 25547 cvmliftlem8 32539 knoppcnlem10 33841 areacirclem2 34998 areacirclem4 35000 |
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