Theorem cnrmi 23369

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 2736	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	restin 23175	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
3		oveq2 7440	. . . 4 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑥) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
4	3	eleq1d 2825	. . 3 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm))
5	1	iscnrm 23332	. . . . 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 4237	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
9		inex1g 5318	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V)
10		elpwg 4602	. . . . . 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 3622	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm)
15	2, 14	eqeltrd 2840	1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ∩ cin 3949   ⊆ wss 3950  𝒫 cpw 4599  ∪ cuni 4906  (class class class)co 7432   ↾_t crest 17466  Topctop 22900  Nrmcnrm 23319  CNrmccnrm 23320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-rest 17468  df-cnrm 23327

This theorem is referenced by:  cnrmnrm  23370  restcnrm  23371