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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 4840 | . 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 12109 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | sseqtrd 3105 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
8 | eqid 2117 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cnrest 12331 | . . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿) ∧ 𝑌 ⊆ ∪ 𝐽) → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
10 | 3, 7, 9 | syl2anc 408 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
11 | cnmpt1res.2 | . . . 4 ⊢ 𝐾 = (𝐽 ↾t 𝑌) | |
12 | 11 | oveq1i 5752 | . . 3 ⊢ (𝐾 Cn 𝐿) = ((𝐽 ↾t 𝑌) Cn 𝐿) |
13 | 10, 12 | eleqtrrdi 2211 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ (𝐾 Cn 𝐿)) |
14 | 2, 13 | eqeltrrd 2195 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 ∪ cuni 3706 ↦ cmpt 3959 ↾ cres 4511 ‘cfv 5093 (class class class)co 5742 ↾t crest 12047 TopOnctopon 12104 Cn ccn 12281 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-map 6512 df-rest 12049 df-topgen 12068 df-top 12092 df-topon 12105 df-bases 12137 df-cn 12284 |
This theorem is referenced by: (None) |
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