Theorem nrmsep3 20969

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 20949	. . . . . 6 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2	1	simprbi 479	. . . . 5 ⊢ (𝐽 ∈ Nrm → ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦))
3		pweq 4111	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4	3	ineq2d 3776	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((Clsd‘𝐽) ∩ 𝒫 𝑦) = ((Clsd‘𝐽) ∩ 𝒫 𝐴))
5		sseq2 3590	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
6	5	anbi2d 736	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
7	6	rexbidv 3034	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
8	4, 7	raleqbidv 3129	. . . . . 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)))
9	8	rspccv 3279	. . . . 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)))
10	2, 9	syl 17	. . . 4 ⊢ (𝐽 ∈ Nrm → (𝐴 ∈ 𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
11		elin 3758	. . . . . 6 ⊢ (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴))
12		elpwg 4116	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
13	12	pm5.32i 667	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴))
14	11, 13	bitri 263	. . . . 5 ⊢ (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴))
15		sseq1 3589	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑥))
16	15	anbi1d 737	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
17	16	rexbidv 3034	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐽 (𝑧 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
18	17	rspccv 3279	. . . . 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)))
19	14, 18	syl5bir 232	. . . 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)))
20	10, 19	syl6 34	. . 3 ⊢ (𝐽 ∈ Nrm → (𝐴 ∈ 𝐽 → ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))))
21	20	exp4a 631	. 2 ⊢ (𝐽 ∈ Nrm → (𝐴 ∈ 𝐽 → (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ⊆ 𝐴 → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))))
22	21	3imp2 1274	1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ‘cfv 5804  Topctop 20517  Clsdccld 20630  clsccl 20632  Nrmcnrm 20924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-nrm 20931

This theorem is referenced by:  nrmsep2  20970  kqnrmlem1  21356  kqnrmlem2  21357  nrmr0reg  21362  nrmhmph  21407