Theorem isnrm 21517

Description: The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
isnrm	⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐽

Proof of Theorem isnrm

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6437	. . . . 5 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2	1	ineq1d 4042	. . . 4 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ∩ 𝒫 𝑥) = ((Clsd‘𝐽) ∩ 𝒫 𝑥))
3		fveq2 6437	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
4	3	fveq1d 6439	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘𝑧) = ((cls‘𝐽)‘𝑧))
5	4	sseq1d 3857	. . . . . 6 ⊢ (𝑗 = 𝐽 → (((cls‘𝑗)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
6	5	anbi2d 622	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
7	6	rexeqbi1dv 3359	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
8	2, 7	raleqbidv 3364	. . 3 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
9	8	raleqbi1dv 3358	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
10		df-nrm 21499	. 2 ⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
11	9, 10	elrab2 3589	1 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118   ∩ cin 3797   ⊆ wss 3798  𝒫 cpw 4380  ‘cfv 6127  Topctop 21075  Clsdccld 21198  clsccl 21200  Nrmcnrm 21492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-nrm 21499

This theorem is referenced by:  nrmtop  21518  nrmsep3  21537  isnrm2  21540  kqnrmlem1  21924  kqnrmlem2  21925  nrmhmph  21975