Theorem isnrm 23364

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 6920	. . . . 5 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2	1	ineq1d 4240	. . . 4 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ∩ 𝒫 𝑥) = ((Clsd‘𝐽) ∩ 𝒫 𝑥))
3		fveq2 6920	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
4	3	fveq1d 6922	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘𝑧) = ((cls‘𝐽)‘𝑧))
5	4	sseq1d 4040	. . . . . 6 ⊢ (𝑗 = 𝐽 → (((cls‘𝑗)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
6	5	anbi2d 629	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
7	6	rexeqbi1dv 3347	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
8	2, 7	raleqbidv 3354	. . 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 3346	. 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 23346	. 2 ⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
11	9, 10	elrab2 3711	1 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ‘cfv 6573  Topctop 22920  Clsdccld 23045  clsccl 23047  Nrmcnrm 23339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-nrm 23346

This theorem is referenced by:  nrmtop  23365  nrmsep3  23384  isnrm2  23387  kqnrmlem1  23772  kqnrmlem2  23773  nrmhmph  23823