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Theorem cldregopn 32021
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 20791 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽))
3 eqcom 2628 . . . . 5 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
43biimpi 206 . . . 4 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
5 fveq2 6158 . . . . . 6 (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
65eqeq2d 2631 . . . . 5 (𝑐 = ((cls‘𝐽)‘𝐴) → (𝐴 = ((int‘𝐽)‘𝑐) ↔ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))))
76rspcev 3299 . . . 4 ((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
82, 4, 7syl2an 494 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
98ex 450 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
10 cldrcl 20770 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
111cldss 20773 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝑐𝑋)
121ntrss2 20801 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
1310, 11, 12syl2anc 692 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
141clsss2 20816 . . . . . . . 8 ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
1513, 14mpdan 701 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
161ntrss 20799 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
1710, 11, 15, 16syl3anc 1323 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
181ntridm 20812 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
1910, 11, 18syl2anc 692 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
201ntrss3 20804 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
2110, 11, 20syl2anc 692 . . . . . . . . 9 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
221clsss3 20803 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
2310, 21, 22syl2anc 692 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
241sscls 20800 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
2510, 21, 24syl2anc 692 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
261ntrss 20799 . . . . . . . 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))))
2710, 23, 25, 26syl3anc 1323 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2819, 27eqsstr3d 3625 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2917, 28eqssd 3605 . . . . 5 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
3029adantl 482 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
31 fveq2 6158 . . . . . 6 (𝐴 = ((int‘𝐽)‘𝑐) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
3231fveq2d 6162 . . . . 5 (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
33 id 22 . . . . 5 (𝐴 = ((int‘𝐽)‘𝑐) → 𝐴 = ((int‘𝐽)‘𝑐))
3432, 33eqeq12d 2636 . . . 4 (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)))
3530, 34syl5ibrcom 237 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
3635rexlimdva 3026 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
379, 36impbid 202 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 196  wa 384   = wceq 1480  wcel 1987  wrex 2909  wss 3560   cuni 4409  cfv 5857  Topctop 20638  Clsdccld 20760  intcnt 20761  clsccl 20762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-top 20639  df-cld 20763  df-ntr 20764  df-cls 20765
This theorem is referenced by: (None)
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