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Mirrors > Home > ILE Home > Th. List > cnmpt21f | 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 |
---|---|
cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmpt21.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
cnmpt21f.f | ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀)) |
Ref | Expression |
---|---|
cnmpt21f | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt21.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | cnmpt21.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
3 | cnmpt21.a | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | |
4 | cnmpt21f.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀)) | |
5 | cntop1 13361 | . . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Top) |
7 | toptopon2 13177 | . . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) | |
8 | 6, 7 | sylib 122 | . 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
9 | eqid 2177 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
10 | eqid 2177 | . . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀 | |
11 | 9, 10 | cnf 13364 | . . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀) |
12 | 4, 11 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀) |
13 | 12 | feqmptd 5565 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧))) |
14 | 13, 4 | eqeltrrd 2255 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀)) |
15 | fveq2 5511 | . 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴)) | |
16 | 1, 2, 3, 8, 14, 15 | cnmpt21 13451 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∪ cuni 3807 ↦ cmpt 4061 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ∈ cmpo 5871 Topctop 13155 TopOnctopon 13168 Cn ccn 13345 ×t ctx 13412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-map 6644 df-topgen 12654 df-top 13156 df-topon 13169 df-bases 13201 df-cn 13348 df-tx 13413 |
This theorem is referenced by: cnmpt22 13454 txhmeo 13479 |
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