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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 4870 | . 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 12185 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | sseqtrd 3135 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
8 | eqid 2139 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cnrest 12407 | . . . 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 5784 | . . 3 ⊢ (𝐾 Cn 𝐿) = ((𝐽 ↾t 𝑌) Cn 𝐿) |
13 | 10, 12 | eleqtrrdi 2233 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ (𝐾 Cn 𝐿)) |
14 | 2, 13 | eqeltrrd 2217 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 ∪ cuni 3736 ↦ cmpt 3989 ↾ cres 4541 ‘cfv 5123 (class class class)co 5774 ↾t crest 12123 TopOnctopon 12180 Cn ccn 12357 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-rest 12125 df-topgen 12144 df-top 12168 df-topon 12181 df-bases 12213 df-cn 12360 |
This theorem is referenced by: (None) |
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