Theorem iscnrm3rlem7 47668

Description: Lemma for iscnrm3rlem8 47669. 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.)

Hypotheses

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)))))

Assertion

Ref	Expression
iscnrm3rlem7	⊢ (𝜑 → 𝑂 ∈ 𝐽)

Proof of Theorem iscnrm3rlem7

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 7735	. . . . . . 7 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
6	5	difexd 5330	. . . . . 6 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V)
7		resttop 22885	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top)
9		eqid 2731	. . . . . 6 ⊢ ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))
10	9	eltopss 22630	. . . . 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 583	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
12		difssd 4133	. . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ ∪ 𝐽)
13		eqid 2731	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
14	13	restuni 22887	. . . . 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 583	. . . 4 ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = ∪ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
16	11, 15	sseqtrrd 4024	. . 3 ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))
17	2, 3, 4, 16	iscnrm3rlem6 47667	. 2 ⊢ (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽))
18	1, 17	mpbid 231	1 ⊢ (𝜑 → 𝑂 ∈ 𝐽)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  ∪ cuni 4909  ‘cfv 6544  (class class class)co 7412   ↾_t crest 17371  Topctop 22616  clsccl 22743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-en 8943  df-fin 8946  df-fi 9409  df-rest 17373  df-topgen 17394  df-top 22617  df-topon 22634  df-bases 22670  df-cld 22744  df-cls 22746

This theorem is referenced by:  iscnrm3rlem8  47669