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Theorem iscnrm3rlem7 46950
Description: Lemma for iscnrm3rlem8 46951. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
Hypotheses
Ref Expression
iscnrm3rlem4.1 (𝜑𝐽 ∈ Top)
iscnrm3rlem4.2 (𝜑𝑆 𝐽)
iscnrm3rlem5.3 (𝜑𝑇 𝐽)
iscnrm3rlem7.4 (𝜑𝑂 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
Assertion
Ref Expression
iscnrm3rlem7 (𝜑𝑂𝐽)

Proof of Theorem iscnrm3rlem7
StepHypRef Expression
1 iscnrm3rlem7.4 . 2 (𝜑𝑂 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
2 iscnrm3rlem4.1 . . 3 (𝜑𝐽 ∈ Top)
3 iscnrm3rlem4.2 . . 3 (𝜑𝑆 𝐽)
4 iscnrm3rlem5.3 . . 3 (𝜑𝑇 𝐽)
52uniexd 7678 . . . . . . 7 (𝜑 𝐽 ∈ V)
65difexd 5286 . . . . . 6 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V)
7 resttop 22509 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V) → (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top)
82, 6, 7syl2anc 584 . . . . 5 (𝜑 → (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top)
9 eqid 2736 . . . . . 6 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))
109eltopss 22254 . . . . 5 (((𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top ∧ 𝑂 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) → 𝑂 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
118, 1, 10syl2anc 584 . . . 4 (𝜑𝑂 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
12 difssd 4092 . . . . 5 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ 𝐽)
13 eqid 2736 . . . . . 6 𝐽 = 𝐽
1413restuni 22511 . . . . 5 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ 𝐽) → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
152, 12, 14syl2anc 584 . . . 4 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
1611, 15sseqtrrd 3985 . . 3 (𝜑𝑂 ⊆ ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))
172, 3, 4, 16iscnrm3rlem6 46949 . 2 (𝜑 → (𝑂 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂𝐽))
181, 17mpbid 231 1 (𝜑𝑂𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  cin 3909  wss 3910   cuni 4865  cfv 6496  (class class class)co 7356  t crest 17301  Topctop 22240  clsccl 22367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7802  df-1st 7920  df-2nd 7921  df-en 8883  df-fin 8886  df-fi 9346  df-rest 17303  df-topgen 17324  df-top 22241  df-topon 22258  df-bases 22294  df-cld 22368  df-cls 22370
This theorem is referenced by:  iscnrm3rlem8  46951
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