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Mirrors > Home > ILE Home > Th. List > cnmpt11f | 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 12749 | . . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Top) |
5 | eqid 2164 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
6 | 5 | toptopon 12563 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
7 | 4, 6 | sylib 121 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
8 | cnmpt11f.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) | |
9 | eqid 2164 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | 5, 9 | cnf 12751 | . . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿) |
11 | 8, 10 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿) |
12 | 11 | feqmptd 5533 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦))) |
13 | 12, 8 | eqeltrrd 2242 | . 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿)) |
14 | fveq2 5480 | . 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
15 | 1, 2, 7, 13, 14 | cnmpt11 12830 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ∪ cuni 3783 ↦ cmpt 4037 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 Topctop 12542 TopOnctopon 12555 Cn ccn 12732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-top 12543 df-topon 12556 df-cn 12735 |
This theorem is referenced by: cnmpt12f 12833 |
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