Theorem restopnb 23069

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 3960	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 ⊆ 𝐴)
4		dfss2 3935	. . . . . 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 2736	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 = (𝐶 ∩ 𝐴))
7		ineq1 4179	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝑣 ∩ 𝐴) = (𝐶 ∩ 𝐴))
8	7	rspceeqv 3614	. . . . 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 4194	. . . . . 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 4185	. . . . . . 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 3935	. . . . . . . . 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 2767	. . . . . 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 4189	. . . . . . . . 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 4186	. . . . . . 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 2789	. . . . 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 22793	. . . . . 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 2829	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 ∈ 𝐽)
31	30	rexlimdvaa 3136	. . 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 17397	. . 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 1540   ∈ wcel 2109  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  (class class class)co 7390   ↾_t crest 17390  Topctop 22787

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-rest 17392  df-top 22788

This theorem is referenced by:  restopn2  23071  cxpcn3  26665  pnfneige0  33948  fourierdlem62  46173  fouriersw  46236  iooii  48910