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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm3rlem7 | Structured version Visualization version GIF version |
Description: Lemma for iscnrm3rlem8 46951. 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 7678 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝐽 ∈ V) |
6 | 5 | difexd 5286 | . . . . . 6 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V) |
7 | resttop 22509 | . . . . . 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 2736 | . . . . . 6 ⊢ ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) | |
10 | 9 | eltopss 22254 | . . . . 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 4092 | . . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ ∪ 𝐽) | |
13 | eqid 2736 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
14 | 13 | restuni 22511 | . . . . 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 3985 | . . 3 ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) |
17 | 2, 3, 4, 16 | iscnrm3rlem6 46949 | . 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 1541 ∈ wcel 2106 Vcvv 3445 ∖ cdif 3907 ∩ cin 3909 ⊆ wss 3910 ∪ cuni 4865 ‘cfv 6496 (class class class)co 7356 ↾t crest 17301 Topctop 22240 clsccl 22367 |
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 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
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 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-en 8883 df-fin 8886 df-fi 9346 df-rest 17303 df-topgen 17324 df-top 22241 df-topon 22258 df-bases 22294 df-cld 22368 df-cls 22370 |
This theorem is referenced by: iscnrm3rlem8 46951 |
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