Theorem cnrmi 23384

Description: A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
cnrmi	⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm)

Proof of Theorem cnrmi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	restin 23190	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
3		oveq2 7439	. . . 4 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑥) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
4	3	eleq1d 2824	. . 3 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm))
5	1	iscnrm 23347	. . . . 5 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
6	5	simprbi 496	. . . 4 ⊢ (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)
7	6	adantr 480	. . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)
8		inss2 4246	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
9		inex1g 5325	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V)
10		elpwg 4608	. . . . . 6 ⊢ ((𝐴 ∩ ∪ 𝐽) ∈ V → ((𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽))
11	9, 10	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽))
12	8, 11	mpbiri 258	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽)
13	12	adantl 481	. . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽)
14	4, 7, 13	rspcdva 3623	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm)
15	2, 14	eqeltrd 2839	1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  (class class class)co 7431   ↾_t crest 17467  Topctop 22915  Nrmcnrm 23334  CNrmccnrm 23335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17469  df-cnrm 23342

This theorem is referenced by:  cnrmnrm  23385  restcnrm  23386