Theorem nrmsep3 23566

Description: In a normal space, given a closed set 𝐵 inside an open set 𝐴, there is an open set 𝑥 such that 𝐵 ⊆ 𝑥 ⊆ cls(𝑥) ⊆ 𝐴. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
nrmsep3	⊢ ((𝐽 ∈ Nrm ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐽

Proof of Theorem nrmsep3

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnrm 23546	. . . . 5 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2		pweq 4578	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
3	2	ineq2d 4173	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((Clsd‘𝐽) ∩ 𝒫 𝑦) = ((Clsd‘𝐽) ∩ 𝒫 𝐴))
4		sseq2 3964	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
5	4	anbi2d 642	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
6	5	rexbidv 3191	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
7	3, 6	raleqbidv 3340	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
8	7	rspccv 3580	. . . . 5 ⊢ (∀𝑦 ∈ 𝐽 ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) → (𝐴 ∈ 𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
9	1, 8	simplbiim 514	. . . 4 ⊢ (𝐽 ∈ Nrm → (𝐴 ∈ 𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
10		elin 3922	. . . . . 6 ⊢ (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴))
11		elpwg 4567	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
12	11	pm5.32i 585	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴))
13	10, 12	bitri 278	. . . . 5 ⊢ (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴))
14		cleq1lem 15047	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
15	14	rexbidv 3191	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
16	15	rspccv 3580	. . . . 5 ⊢ (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) → (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
17	13, 16	biimtrrid 246	. . . 4 ⊢ (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) → ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
18	9, 17	syl6 36	. . 3 ⊢ (𝐽 ∈ Nrm → (𝐴 ∈ 𝐽 → ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))))
19	18	exp4a 437	. 2 ⊢ (𝐽 ∈ Nrm → (𝐴 ∈ 𝐽 → (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ⊆ 𝐴 → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))))
20	19	3imp2 1368	1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564  ‘cfv 6540  Topctop 23104  Clsdccld 23227  clsccl 23229  Nrmcnrm 23521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-nrm 23528

This theorem is used by:  nrmsep2  23567  kqnrmlem1  23955  kqnrmlem2  23956  nrmr0reg  23961  nrmhmph  24006