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Theorem restcls2 47817
Description: A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
Hypotheses
Ref Expression
restcls2.1 (𝜑𝐽 ∈ Top)
restcls2.2 (𝜑𝑋 = 𝐽)
restcls2.3 (𝜑𝑌𝑋)
restcls2.4 (𝜑𝐾 = (𝐽t 𝑌))
restcls2.5 (𝜑𝑆 ∈ (Clsd‘𝐾))
Assertion
Ref Expression
restcls2 (𝜑𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restcls2
StepHypRef Expression
1 restcls2.4 . . . 4 (𝜑𝐾 = (𝐽t 𝑌))
21fveq2d 6889 . . 3 (𝜑 → (cls‘𝐾) = (cls‘(𝐽t 𝑌)))
32fveq1d 6887 . 2 (𝜑 → ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽t 𝑌))‘𝑆))
4 restcls2.5 . . 3 (𝜑𝑆 ∈ (Clsd‘𝐾))
5 cldcls 22901 . . 3 (𝑆 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑆) = 𝑆)
64, 5syl 17 . 2 (𝜑 → ((cls‘𝐾)‘𝑆) = 𝑆)
7 restcls2.1 . . 3 (𝜑𝐽 ∈ Top)
8 restcls2.3 . . . 4 (𝜑𝑌𝑋)
9 restcls2.2 . . . 4 (𝜑𝑋 = 𝐽)
108, 9sseqtrd 4017 . . 3 (𝜑𝑌 𝐽)
117, 9, 8, 1, 4restcls2lem 47816 . . 3 (𝜑𝑆𝑌)
12 eqid 2726 . . . 4 𝐽 = 𝐽
13 eqid 2726 . . . 4 (𝐽t 𝑌) = (𝐽t 𝑌)
1412, 13restcls 23040 . . 3 ((𝐽 ∈ Top ∧ 𝑌 𝐽𝑆𝑌) → ((cls‘(𝐽t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
157, 10, 11, 14syl3anc 1368 . 2 (𝜑 → ((cls‘(𝐽t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
163, 6, 153eqtr3d 2774 1 (𝜑𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cin 3942  wss 3943   cuni 4902  cfv 6537  (class class class)co 7405  t crest 17375  Topctop 22750  Clsdccld 22875  clsccl 22877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17377  df-topgen 17398  df-top 22751  df-topon 22768  df-bases 22804  df-cld 22878  df-cls 22880
This theorem is referenced by:  restclsseplem  47818
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