Theorem restopnb 23119

Description: If 𝐵 is an open subset of the subspace base set 𝐴, then any subset of 𝐵 is open iff it is open in 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
restopnb	⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 ↔ 𝐶 ∈ (𝐽 ↾t 𝐴)))

Proof of Theorem restopnb

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 1197	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 ⊆ 𝐵)
2		simpr2 1196	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐵 ⊆ 𝐴)
3	1, 2	sstrd 3944	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 ⊆ 𝐴)
4		dfss2 3919	. . . . . 6 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶)
5	3, 4	sylib 218	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∩ 𝐴) = 𝐶)
6	5	eqcomd 2742	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 = (𝐶 ∩ 𝐴))
7		ineq1 4165	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝑣 ∩ 𝐴) = (𝐶 ∩ 𝐴))
8	7	rspceeqv 3599	. . . . 5 ⊢ ((𝐶 ∈ 𝐽 ∧ 𝐶 = (𝐶 ∩ 𝐴)) → ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴))
9	8	expcom 413	. . . 4 ⊢ (𝐶 = (𝐶 ∩ 𝐴) → (𝐶 ∈ 𝐽 → ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
10	6, 9	syl 17	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 → ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
11		inass 4180	. . . . . 6 ⊢ ((𝑣 ∩ 𝐴) ∩ 𝐵) = (𝑣 ∩ (𝐴 ∩ 𝐵))
12		simprr 772	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 = (𝑣 ∩ 𝐴))
13	12	ineq1d 4171	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝐶 ∩ 𝐵) = ((𝑣 ∩ 𝐴) ∩ 𝐵))
14		simplr3 1218	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → 𝐶 ⊆ 𝐵)
15		dfss2 3919	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐶 ∩ 𝐵) = 𝐶)
16	14, 15	sylib 218	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → (𝐶 ∩ 𝐵) = 𝐶)
17	16	adantrr 717	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝐶 ∩ 𝐵) = 𝐶)
18	13, 17	eqtr3d 2773	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → ((𝑣 ∩ 𝐴) ∩ 𝐵) = 𝐶)
19		simplr2 1217	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → 𝐵 ⊆ 𝐴)
20		sseqin2 4175	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
21	19, 20	sylib 218	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → (𝐴 ∩ 𝐵) = 𝐵)
22	21	ineq2d 4172	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → (𝑣 ∩ (𝐴 ∩ 𝐵)) = (𝑣 ∩ 𝐵))
23	22	adantrr 717	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝑣 ∩ (𝐴 ∩ 𝐵)) = (𝑣 ∩ 𝐵))
24	11, 18, 23	3eqtr3a 2795	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 = (𝑣 ∩ 𝐵))
25		simplll 774	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐽 ∈ Top)
26		simprl 770	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝑣 ∈ 𝐽)
27		simplr1 1216	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐵 ∈ 𝐽)
28		inopn 22843	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝑣 ∩ 𝐵) ∈ 𝐽)
29	25, 26, 27, 28	syl3anc 1373	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝑣 ∩ 𝐵) ∈ 𝐽)
30	24, 29	eqeltrd 2836	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 ∈ 𝐽)
31	30	rexlimdvaa 3138	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴) → 𝐶 ∈ 𝐽))
32	10, 31	impbid 212	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 ↔ ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
33		elrest 17347	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐶 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
34	33	adantr 480	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
35	32, 34	bitr4d 282	1 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 ↔ 𝐶 ∈ (𝐽 ↾t 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  (class class class)co 7358   ↾_t crest 17340  Topctop 22837

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-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  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-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-rest 17342  df-top 22838

This theorem is referenced by:  restopn2  23121  cxpcn3  26714  pnfneige0  34108  fourierdlem62  46412  fouriersw  46475  iooii  49163