Theorem iscnrm2 23280

Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
iscnrm2	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem iscnrm2

Step	Hyp	Ref	Expression
1		topontop 22855	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		eqid 2734	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	iscnrm 23265	. . . 4 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
4	3	baib 535	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
5	1, 4	syl 17	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
6		toponuni 22856	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	pweqd 4569	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
8	7	raleqdv 3294	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
9	5, 8	bitr4d 282	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  ∀wral 3049  𝒫 cpw 4552  ∪ cuni 4861  ‘cfv 6490  (class class class)co 7356   ↾_t crest 17338  Topctop 22835  TopOnctopon 22852  Nrmcnrm 23252  CNrmccnrm 23253

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-topon 22853  df-cnrm 23260

This theorem is referenced by:  restcnrm  23304