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Mirrors > Home > ILE Home > Th. List > stoig | GIF version |
Description: The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
restuni.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
stoig | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), (𝐽 ↾t 𝐴)〉} ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | toptopon 12563 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | resttopon 12718 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
4 | 2, 3 | sylanb 282 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
5 | eqid 2164 | . . 3 ⊢ {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), (𝐽 ↾t 𝐴)〉} = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), (𝐽 ↾t 𝐴)〉} | |
6 | 5 | eltpsg 12585 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), (𝐽 ↾t 𝐴)〉} ∈ TopSp) |
7 | 4, 6 | syl 14 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), (𝐽 ↾t 𝐴)〉} ∈ TopSp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 {cpr 3571 〈cop 3573 ∪ cuni 3783 ‘cfv 5182 (class class class)co 5836 ndxcnx 12334 Basecbs 12337 TopSetcts 12405 ↾t crest 12498 Topctop 12542 TopOnctopon 12555 TopSpctps 12575 |
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 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
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-nel 2430 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-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 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-pnf 7926 df-mnf 7927 df-ltxr 7929 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-tset 12418 df-rest 12500 df-topn 12501 df-topgen 12519 df-top 12543 df-topon 12556 df-topsp 12576 df-bases 12588 |
This theorem is referenced by: (None) |
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