Theorem restcnrm 23306

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 2736	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	restin 23110	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
3		simpll 766	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝐽 ∈ CNrm)
4		elpwi 4561	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽))
5	4	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽))
6		inex1g 5264	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V)
7	6	ad2antlr 727	. . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐴 ∩ ∪ 𝐽) ∈ V)
8		restabs 23109	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ∈ V) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥))
9	3, 5, 7, 8	syl3anc 1373	. . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥))
10		cnrmi 23304	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm)
11	10	adantlr 715	. . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm)
12	9, 11	eqeltrd 2836	. . . 4 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)
13	12	ralrimiva 3128	. . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)
14		cnrmtop 23281	. . . . . . 7 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)
15	14	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ Top)
16		toptopon2 22862	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
17	15, 16	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
18		inss2 4190	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
19		resttopon 23105	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)))
20	17, 18, 19	sylancl 586	. . . 4 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)))
21		iscnrm2 23282	. . . 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 257	. 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm)
24	2, 23	eqeltrd 2836	1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863  ‘cfv 6492  (class class class)co 7358   ↾_t crest 17340  Topctop 22837  TopOnctopon 22854  Nrmcnrm 23254  CNrmccnrm 23255

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17342  df-topgen 17363  df-top 22838  df-topon 22855  df-bases 22890  df-cnrm 23262

This theorem is referenced by: (None)