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Theorem iscnrm3rlem3 47529
Description: Lemma for iscnrm3r 47535. 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 7727 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
2 difssd 4132 . . . 4 (𝐽 ∈ Top → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ 𝐽)
31, 2sselpwd 5326 . . 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 770 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑆 ∈ 𝒫 𝐽)
76elpwid 4611 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑆 𝐽)
85, 7iscnrm3rlem2 47528 . 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 772 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑇 ∈ 𝒫 𝐽)
109elpwid 4611 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → 𝑇 𝐽)
115, 10iscnrm3rlem2 47528 . . 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 4201 . . . . . 6 (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) = (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))
1312difeq2i 4119 . . . . 5 ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)))
1413oveq2i 7417 . . . 4 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))))
1514fveq2i 6892 . . 3 (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) = (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)))))
1611, 15eleqtrrdi 2845 . 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 1129 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 1088  wcel 2107  Vcvv 3475  cdif 3945  cin 3947  𝒫 cpw 4602   cuni 4908  cfv 6541  (class class class)co 7406  t crest 17363  Topctop 22387  Clsdccld 22512  clsccl 22514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-en 8937  df-fin 8940  df-fi 9403  df-rest 17365  df-topgen 17386  df-top 22388  df-topon 22405  df-bases 22441  df-cld 22515  df-cls 22517
This theorem is referenced by:  iscnrm3r  47535
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