Theorem cldregopn 36532

Description: A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)

Hypothesis

Ref	Expression
opnregcld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldregopn	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))

Distinct variable groups:   𝐴,𝑐   𝐽,𝑐   𝑋,𝑐

Proof of Theorem cldregopn

Step	Hyp	Ref	Expression
1		opnregcld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	clscld 23025	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽))
3		eqcom 2744	. . . . 5 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
4	3	biimpi 216	. . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
5		fveq2 6835	. . . . 5 ⊢ (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
6	5	rspceeqv 3588	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
7	2, 4, 6	syl2an 597	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
8	7	ex 412	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
9		cldrcl 23004	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
10	1	cldss 23007	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ 𝑋)
11	1	ntrss2 23035	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
12	9, 10, 11	syl2anc 585	. . . . . . . 8 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
13	1	clsss2 23050	. . . . . . . 8 ⊢ ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
14	12, 13	mpdan 688	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
15	1	ntrss 23033	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
16	9, 10, 14, 15	syl3anc 1374	. . . . . 6 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
17	1	ntridm 23046	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
18	9, 10, 17	syl2anc 585	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
19	1	ntrss3 23038	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
20	9, 10, 19	syl2anc 585	. . . . . . . . 9 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
21	1	clsss3 23037	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
22	9, 20, 21	syl2anc 585	. . . . . . . 8 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
23	1	sscls 23034	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
24	9, 20, 23	syl2anc 585	. . . . . . . 8 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
25	1	ntrss 23033	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐))) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
26	9, 22, 24, 25	syl3anc 1374	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
27	18, 26	eqsstrrd 3958	. . . . . 6 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
28	16, 27	eqssd 3940	. . . . 5 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
29	28	adantl 481	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
30		2fveq3 6840	. . . . 5 ⊢ (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
31		id 22	. . . . 5 ⊢ (𝐴 = ((int‘𝐽)‘𝑐) → 𝐴 = ((int‘𝐽)‘𝑐))
32	30, 31	eqeq12d 2753	. . . 4 ⊢ (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)))
33	29, 32	syl5ibrcom 247	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
34	33	rexlimdva 3139	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
35	8, 34	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890  ∪ cuni 4851  ‘cfv 6493  Topctop 22871  Clsdccld 22994  intcnt 22995  clsccl 22996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22872  df-cld 22997  df-ntr 22998  df-cls 22999

This theorem is referenced by: (None)