Theorem restcnrm 21544

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 2825	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	restin 21348	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
3		simpll 783	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝐽 ∈ CNrm)
4		elpwi 4390	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽))
5	4	adantl 475	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽))
6		inex1g 5028	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V)
7	6	ad2antlr 718	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐴 ∩ ∪ 𝐽) ∈ V)
8		restabs 21347	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ∈ V) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥))
9	3, 5, 7, 8	syl3anc 1494	. . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥))
10		cnrmi 21542	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm)
11	10	adantlr 706	. . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm)
12	9, 11	eqeltrd 2906	. . . 4 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)
13	12	ralrimiva 3175	. . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)
14		cnrmtop 21519	. . . . . . 7 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)
15	14	adantr 474	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ Top)
16	1	toptopon 21099	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
17	15, 16	sylib 210	. . . . 5 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
18		inss2 4060	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
19		resttopon 21343	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)))
20	17, 18, 19	sylancl 580	. . . 4 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)))
21		iscnrm2 21520	. . . 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 249	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm)
24	2, 23	eqeltrd 2906	1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ∩ cin 3797   ⊆ wss 3798  𝒫 cpw 4380  ∪ cuni 4660  ‘cfv 6127  (class class class)co 6910   ↾_t crest 16441  Topctop 21075  TopOnctopon 21092  Nrmcnrm 21492  CNrmccnrm 21493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-oadd 7835  df-er 8014  df-en 8229  df-fin 8232  df-fi 8592  df-rest 16443  df-topgen 16464  df-top 21076  df-topon 21093  df-bases 21128  df-cnrm 21500

This theorem is referenced by: (None)