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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm3rlem7 | Structured version Visualization version GIF version |
Description: Lemma for iscnrm3rlem8 46241. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.) |
Ref | Expression |
---|---|
iscnrm3rlem4.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
iscnrm3rlem4.2 | ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
iscnrm3rlem5.3 | ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) |
iscnrm3rlem7.4 | ⊢ (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) |
Ref | Expression |
---|---|
iscnrm3rlem7 | ⊢ (𝜑 → 𝑂 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscnrm3rlem7.4 | . 2 ⊢ (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) | |
2 | iscnrm3rlem4.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | iscnrm3rlem4.2 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) | |
4 | iscnrm3rlem5.3 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) | |
5 | 2 | uniexd 7595 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝐽 ∈ V) |
6 | 5 | difexd 5253 | . . . . . 6 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V) |
7 | resttop 22311 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V) → (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top) | |
8 | 2, 6, 7 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top) |
9 | eqid 2738 | . . . . . 6 ⊢ ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) | |
10 | 9 | eltopss 22056 | . . . . 5 ⊢ (((𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top ∧ 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) → 𝑂 ⊆ ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) |
11 | 8, 1, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑂 ⊆ ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) |
12 | difssd 4067 | . . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ ∪ 𝐽) | |
13 | eqid 2738 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
14 | 13 | restuni 22313 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ ∪ 𝐽) → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) |
15 | 2, 12, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) |
16 | 11, 15 | sseqtrrd 3962 | . . 3 ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) |
17 | 2, 3, 4, 16 | iscnrm3rlem6 46239 | . 2 ⊢ (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽)) |
18 | 1, 17 | mpbid 231 | 1 ⊢ (𝜑 → 𝑂 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 ∪ cuni 4839 ‘cfv 6433 (class class class)co 7275 ↾t crest 17131 Topctop 22042 clsccl 22169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-en 8734 df-fin 8737 df-fi 9170 df-rest 17133 df-topgen 17154 df-top 22043 df-topon 22060 df-bases 22096 df-cld 22170 df-cls 22172 |
This theorem is referenced by: iscnrm3rlem8 46241 |
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