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Theorem iscnrm3rlem7 45762
Description: Lemma for iscnrm3rlem8 45763. 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 7486 . . . . . . 7 (𝜑 𝐽 ∈ V)
65difexd 5197 . . . . . 6 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V)
7 resttop 21911 . . . . . 6 ((𝐽 ∈ Top ∧ ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ V) → (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top)
82, 6, 7syl2anc 587 . . . . 5 (𝜑 → (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∈ Top)
9 eqid 2738 . . . . . 6 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))
109eltopss 21658 . . . . 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 587 . . . 4 (𝜑𝑂 (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
12 difssd 4023 . . . . 5 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ 𝐽)
13 eqid 2738 . . . . . 6 𝐽 = 𝐽
1413restuni 21913 . . . . 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 587 . . . 4 (𝜑 → ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
1611, 15sseqtrrd 3918 . . 3 (𝜑𝑂 ⊆ ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))
172, 3, 4, 16iscnrm3rlem6 45761 . 2 (𝜑 → (𝑂 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂𝐽))
181, 17mpbid 235 1 (𝜑𝑂𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  Vcvv 3398  cdif 3840  cin 3842  wss 3843   cuni 4796  cfv 6339  (class class class)co 7170  t crest 16797  Topctop 21644  clsccl 21769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-en 8556  df-fin 8559  df-fi 8948  df-rest 16799  df-topgen 16820  df-top 21645  df-topon 21662  df-bases 21697  df-cld 21770  df-cls 21772
This theorem is referenced by:  iscnrm3rlem8  45763
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