Theorem restcnrm 21972

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

Assertion

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

Proof of Theorem restcnrm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	restin 21776	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
3		simpll 765	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝐽 ∈ CNrm)
4		elpwi 4550	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽))
5	4	adantl 484	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽))
6		inex1g 5225	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V)
7	6	ad2antlr 725	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐴 ∩ ∪ 𝐽) ∈ V)
8		restabs 21775	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ∈ V) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥))
9	3, 5, 7, 8	syl3anc 1367	. . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥))
10		cnrmi 21970	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm)
11	10	adantlr 713	. . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm)
12	9, 11	eqeltrd 2915	. . . 4 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)
13	12	ralrimiva 3184	. . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)
14		cnrmtop 21947	. . . . . . 7 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)
15	14	adantr 483	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ Top)
16		toptopon2 21528	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
17	15, 16	sylib 220	. . . . 5 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
18		inss2 4208	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
19		resttopon 21771	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)))
20	17, 18, 19	sylancl 588	. . . 4 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)))
21		iscnrm2 21948	. . . 4 ⊢ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm))
22	20, 21	syl 17	. . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm))
23	13, 22	mpbird 259	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm)
24	2, 23	eqeltrd 2915	1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840  ‘cfv 6357  (class class class)co 7158   ↾_t crest 16696  Topctop 21503  TopOnctopon 21520  Nrmcnrm 21920  CNrmccnrm 21921

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-oadd 8108  df-er 8291  df-en 8512  df-fin 8515  df-fi 8877  df-rest 16698  df-topgen 16719  df-top 21504  df-topon 21521  df-bases 21556  df-cnrm 21928

This theorem is referenced by: (None)