Theorem cnrmi 23280

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 2729	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	restin 23086	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
3		oveq2 7377	. . . 4 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑥) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
4	3	eleq1d 2813	. . 3 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm))
5	1	iscnrm 23243	. . . . 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 4197	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
9		inex1g 5269	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V)
10		elpwg 4562	. . . . . 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 3586	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm)
15	2, 14	eqeltrd 2828	1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867  (class class class)co 7369   ↾_t crest 17359  Topctop 22813  Nrmcnrm 23230  CNrmccnrm 23231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-rest 17361  df-cnrm 23238

This theorem is referenced by:  cnrmnrm  23281  restcnrm  23282