![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > resttopon2 | Structured version Visualization version GIF version |
Description: The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
resttopon2 | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ (TopOn‘(𝐴 ∩ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22246 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | resttop 22495 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top) |
4 | toponuni 22247 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
5 | 4 | ineq2d 4170 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∩ 𝑋) = (𝐴 ∩ ∪ 𝐽)) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝑋) = (𝐴 ∩ ∪ 𝐽)) |
7 | eqid 2736 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7 | restuni2 22502 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) = ∪ (𝐽 ↾t 𝐴)) |
9 | 1, 8 | sylan 580 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) = ∪ (𝐽 ↾t 𝐴)) |
10 | 6, 9 | eqtrd 2776 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t 𝐴)) |
11 | istopon 22245 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘(𝐴 ∩ 𝑋)) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t 𝐴))) | |
12 | 3, 10, 11 | sylanbrc 583 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ (TopOn‘(𝐴 ∩ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3907 ∪ cuni 4863 ‘cfv 6493 (class class class)co 7353 ↾t crest 17294 Topctop 22226 TopOnctopon 22243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-en 8880 df-fin 8883 df-fi 9343 df-rest 17296 df-topgen 17317 df-top 22227 df-topon 22244 df-bases 22280 |
This theorem is referenced by: resstps 22522 lmss 22633 |
Copyright terms: Public domain | W3C validator |