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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 21423 | . . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Top) |
5 | eqid 2825 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
6 | 5 | toptopon 21099 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
7 | 4, 6 | sylib 210 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
8 | cnmpt11f.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) | |
9 | eqid 2825 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | 5, 9 | cnf 21428 | . . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿) |
12 | 11 | feqmptd 6500 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦))) |
13 | 12, 8 | eqeltrrd 2907 | . 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿)) |
14 | fveq2 6437 | . 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
15 | 1, 2, 7, 13, 14 | cnmpt11 21844 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ∪ cuni 4660 ↦ cmpt 4954 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 Topctop 21075 TopOnctopon 21092 Cn ccn 21406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 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 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-top 21076 df-topon 21093 df-cn 21409 |
This theorem is referenced by: cnmpt12f 21847 tgpmulg 22274 prdstgpd 22305 pcorevcl 23201 pcorevlem 23202 logcn 24799 loglesqrt 24908 efrlim 25116 cvmliftlem8 31816 knoppcnlem10 33020 areacirclem2 34039 areacirclem4 34041 |
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