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Theorem opnregcld 32631
Description: A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
Hypothesis
Ref Expression
opnregcld.1 𝑋 = 𝐽
Assertion
Ref Expression
opnregcld ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
Distinct variable groups:   𝐴,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem opnregcld
StepHypRef Expression
1 opnregcld.1 . . . . 5 𝑋 = 𝐽
21ntropn 21055 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3 eqcom 2767 . . . . 5 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
43biimpi 206 . . . 4 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5 fveq2 6352 . . . . . 6 (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
65eqeq2d 2770 . . . . 5 (𝑜 = ((int‘𝐽)‘𝐴) → (𝐴 = ((cls‘𝐽)‘𝑜) ↔ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))))
76rspcev 3449 . . . 4 ((((int‘𝐽)‘𝐴) ∈ 𝐽𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
82, 4, 7syl2an 495 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
98ex 449 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
10 simpl 474 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝐽 ∈ Top)
111eltopss 20914 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
121clsss3 21065 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
1311, 12syldan 488 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
141ntrss2 21063 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
1513, 14syldan 488 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
161clsss 21060 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜)) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
1710, 13, 15, 16syl3anc 1477 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
181clsidm 21073 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1911, 18syldan 488 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
2017, 19sseqtrd 3782 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
211ntrss3 21066 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
2213, 21syldan 488 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
23 simpr 479 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝐽)
241sscls 21062 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
2511, 24syldan 488 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
261ssntr 21064 . . . . . . . 8 (((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ ((cls‘𝐽)‘𝑜))) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
2710, 13, 23, 25, 26syl22anc 1478 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
281clsss 21060 . . . . . . 7 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜))) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
2910, 22, 27, 28syl3anc 1477 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
3020, 29eqssd 3761 . . . . 5 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
3130adantlr 753 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
32 fveq2 6352 . . . . . 6 (𝐴 = ((cls‘𝐽)‘𝑜) → ((int‘𝐽)‘𝐴) = ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
3332fveq2d 6356 . . . . 5 (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
34 id 22 . . . . 5 (𝐴 = ((cls‘𝐽)‘𝑜) → 𝐴 = ((cls‘𝐽)‘𝑜))
3533, 34eqeq12d 2775 . . . 4 (𝐴 = ((cls‘𝐽)‘𝑜) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜)))
3631, 35syl5ibrcom 237 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
3736rexlimdva 3169 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
389, 37impbid 202 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  wss 3715   cuni 4588  cfv 6049  Topctop 20900  intcnt 21023  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-ntr 21026  df-cls 21027
This theorem is referenced by: (None)
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