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Mirrors > Home > ILE Home > Th. List > cnmpt1res | GIF version |
Description: The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
cnmpt1res.2 | ⊢ 𝐾 = (𝐽 ↾t 𝑌) |
cnmpt1res.3 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt1res.5 | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
cnmpt1res.6 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) |
Ref | Expression |
---|---|
cnmpt1res | ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1res.5 | . . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
2 | 1 | resmptd 4929 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴)) |
3 | cnmpt1res.6 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) | |
4 | cnmpt1res.3 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
5 | toponuni 12560 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | sseqtrd 3175 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
8 | eqid 2164 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cnrest 12782 | . . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿) ∧ 𝑌 ⊆ ∪ 𝐽) → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
10 | 3, 7, 9 | syl2anc 409 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
11 | cnmpt1res.2 | . . . 4 ⊢ 𝐾 = (𝐽 ↾t 𝑌) | |
12 | 11 | oveq1i 5846 | . . 3 ⊢ (𝐾 Cn 𝐿) = ((𝐽 ↾t 𝑌) Cn 𝐿) |
13 | 10, 12 | eleqtrrdi 2258 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ (𝐾 Cn 𝐿)) |
14 | 2, 13 | eqeltrrd 2242 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 ∪ cuni 3783 ↦ cmpt 4037 ↾ cres 4600 ‘cfv 5182 (class class class)co 5836 ↾t crest 12498 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-coll 4091 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-reu 2449 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-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-rest 12500 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 df-cn 12735 |
This theorem is referenced by: (None) |
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