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Theorem cldregopn 33200
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 ∧ 𝐴𝑋) → (((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
StepHypRef Expression
1 opnregcld.1 . . . . 5 𝑋 = 𝐽
21clscld 21353 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽))
3 eqcom 2779 . . . . 5 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
43biimpi 208 . . . 4 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
5 fveq2 6493 . . . . 5 (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
65rspceeqv 3547 . . . 4 ((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
72, 4, 6syl2an 586 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
87ex 405 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
9 cldrcl 21332 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
101cldss 21335 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝑐𝑋)
111ntrss2 21363 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
129, 10, 11syl2anc 576 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
131clsss2 21378 . . . . . . . 8 ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
1412, 13mpdan 674 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
151ntrss 21361 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
169, 10, 14, 15syl3anc 1351 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
171ntridm 21374 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
189, 10, 17syl2anc 576 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
191ntrss3 21366 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
209, 10, 19syl2anc 576 . . . . . . . . 9 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
211clsss3 21365 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
229, 20, 21syl2anc 576 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
231sscls 21362 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
249, 20, 23syl2anc 576 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
251ntrss 21361 . . . . . . . 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))))
269, 22, 24, 25syl3anc 1351 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2718, 26eqsstr3d 3890 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2816, 27eqssd 3869 . . . . 5 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
2928adantl 474 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
30 2fveq3 6498 . . . . 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))
3230, 31eqeq12d 2787 . . . 4 (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)))
3329, 32syl5ibrcom 239 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
3433rexlimdva 3223 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
358, 34impbid 204 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3083  wss 3823   cuni 4706  cfv 6182  Topctop 21199  Clsdccld 21322  intcnt 21323  clsccl 21324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-top 21200  df-cld 21325  df-ntr 21326  df-cls 21327
This theorem is referenced by: (None)
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