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Mirrors > Home > ILE Home > Th. List > restuni2 | GIF version |
Description: The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
restin.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
restuni2 | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ Top) | |
2 | inss2 3338 | . . 3 ⊢ (𝐴 ∩ 𝑋) ⊆ 𝑋 | |
3 | restin.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | restuni 12719 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∩ 𝑋) ⊆ 𝑋) → (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t (𝐴 ∩ 𝑋))) |
5 | 1, 2, 4 | sylancl 410 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t (𝐴 ∩ 𝑋))) |
6 | 3 | restin 12723 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ 𝑋))) |
7 | 6 | unieqd 3794 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t (𝐴 ∩ 𝑋))) |
8 | 5, 7 | eqtr4d 2200 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∩ cin 3110 ⊆ wss 3111 ∪ cuni 3783 (class class class)co 5836 ↾t crest 12498 Topctop 12542 |
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-rest 12500 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 |
This theorem is referenced by: resttopon2 12725 |
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