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Theorem iscnrm3rlem3 46294
Description: Lemma for iscnrm3r 46300. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
iscnrm3rlem3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → (( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))

Proof of Theorem iscnrm3rlem3
StepHypRef Expression
1 uniexg 7625 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
2 difssd 4073 . . . 4 (𝐽 ∈ Top → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ 𝐽)
31, 2sselpwd 5259 . . 3 (𝐽 ∈ Top → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽)
43adantr 482 . 2 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽)
5 simpl 484 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝐽 ∈ Top)
6 simprl 769 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑆 ∈ 𝒫 𝐽)
76elpwid 4548 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑆 𝐽)
85, 7iscnrm3rlem2 46293 . 2 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))
9 simprr 771 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑇 ∈ 𝒫 𝐽)
109elpwid 4548 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑇 𝐽)
115, 10iscnrm3rlem2 46293 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))))))
12 incom 4141 . . . . . 6 (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) = (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))
1312difeq2i 4060 . . . . 5 ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)))
1413oveq2i 7318 . . . 4 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))))
1514fveq2i 6807 . . 3 (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) = (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)))))
1611, 15eleqtrrdi 2848 . 2 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))
174, 8, 163jca 1128 1 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → (( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1087  wcel 2104  Vcvv 3437  cdif 3889  cin 3891  𝒫 cpw 4539   cuni 4844  cfv 6458  (class class class)co 7307  t crest 17176  Topctop 22087  Clsdccld 22212  clsccl 22214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-iin 4934  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-en 8765  df-fin 8768  df-fi 9214  df-rest 17178  df-topgen 17199  df-top 22088  df-topon 22105  df-bases 22141  df-cld 22215  df-cls 22217
This theorem is referenced by:  iscnrm3r  46300
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