Theorem restopnb 21785

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 1192	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 ⊆ 𝐵)
2		simpr2 1191	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐵 ⊆ 𝐴)
3	1, 2	sstrd 3979	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 ⊆ 𝐴)
4		df-ss 3954	. . . . . 6 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶)
5	3, 4	sylib 220	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∩ 𝐴) = 𝐶)
6	5	eqcomd 2829	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐶 = (𝐶 ∩ 𝐴))
7		ineq1 4183	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝑣 ∩ 𝐴) = (𝐶 ∩ 𝐴))
8	7	rspceeqv 3640	. . . . 5 ⊢ ((𝐶 ∈ 𝐽 ∧ 𝐶 = (𝐶 ∩ 𝐴)) → ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴))
9	8	expcom 416	. . . 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 4198	. . . . . 6 ⊢ ((𝑣 ∩ 𝐴) ∩ 𝐵) = (𝑣 ∩ (𝐴 ∩ 𝐵))
12		simprr 771	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 = (𝑣 ∩ 𝐴))
13	12	ineq1d 4190	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝐶 ∩ 𝐵) = ((𝑣 ∩ 𝐴) ∩ 𝐵))
14		simplr3 1213	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → 𝐶 ⊆ 𝐵)
15		df-ss 3954	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐶 ∩ 𝐵) = 𝐶)
16	14, 15	sylib 220	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → (𝐶 ∩ 𝐵) = 𝐶)
17	16	adantrr 715	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝐶 ∩ 𝐵) = 𝐶)
18	13, 17	eqtr3d 2860	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → ((𝑣 ∩ 𝐴) ∩ 𝐵) = 𝐶)
19		simplr2 1212	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → 𝐵 ⊆ 𝐴)
20		sseqin2 4194	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
21	19, 20	sylib 220	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → (𝐴 ∩ 𝐵) = 𝐵)
22	21	ineq2d 4191	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ 𝑣 ∈ 𝐽) → (𝑣 ∩ (𝐴 ∩ 𝐵)) = (𝑣 ∩ 𝐵))
23	22	adantrr 715	. . . . . 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 2882	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 = (𝑣 ∩ 𝐵))
25		simplll 773	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐽 ∈ Top)
26		simprl 769	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝑣 ∈ 𝐽)
27		simplr1 1211	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐵 ∈ 𝐽)
28		inopn 21509	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝑣 ∩ 𝐵) ∈ 𝐽)
29	25, 26, 27, 28	syl3anc 1367	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → (𝑣 ∩ 𝐵) ∈ 𝐽)
30	24, 29	eqeltrd 2915	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) ∧ (𝑣 ∈ 𝐽 ∧ 𝐶 = (𝑣 ∩ 𝐴))) → 𝐶 ∈ 𝐽)
31	30	rexlimdvaa 3287	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴) → 𝐶 ∈ 𝐽))
32	10, 31	impbid 214	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 ↔ ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
33		elrest 16703	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐶 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
34	33	adantr 483	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐽 𝐶 = (𝑣 ∩ 𝐴)))
35	32, 34	bitr4d 284	1 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 ↔ 𝐶 ∈ (𝐽 ↾t 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ∩ cin 3937   ⊆ wss 3938  (class class class)co 7158   ↾_t crest 16696  Topctop 21503

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-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  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-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  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-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-rest 16698  df-top 21504

This theorem is referenced by:  restopn2  21787  cxpcn3  25331  pnfneige0  31196  fourierdlem62  42460  fouriersw  42523