Theorem restcls2 47923

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	⊢ (𝜑 → 𝐽 ∈ Top)
restcls2.2	⊢ (𝜑 → 𝑋 = ∪ 𝐽)
restcls2.3	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
restcls2.4	⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))
restcls2.5	⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))

Assertion

Ref	Expression
restcls2	⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restcls2

Step	Hyp	Ref	Expression
1		restcls2.4	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))
2	1	fveq2d 6896	. . 3 ⊢ (𝜑 → (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌)))
3	2	fveq1d 6894	. 2 ⊢ (𝜑 → ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆))
4		restcls2.5	. . 3 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
5		cldcls 22940	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑆) = 𝑆)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ((cls‘𝐾)‘𝑆) = 𝑆)
7		restcls2.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
8		restcls2.3	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
9		restcls2.2	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
10	8, 9	sseqtrd 4019	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽)
11	7, 9, 8, 1, 4	restcls2lem 47922	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12		eqid 2728	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
13		eqid 2728	. . . 4 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌)
14	12, 13	restcls 23079	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
15	7, 10, 11, 14	syl3anc 1369	. 2 ⊢ (𝜑 → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
16	3, 6, 15	3eqtr3d 2776	1 ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ∩ cin 3944   ⊆ wss 3945  ∪ cuni 4904  ‘cfv 6543  (class class class)co 7415   ↾_t crest 17396  Topctop 22789  Clsdccld 22914  clsccl 22916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-en 8959  df-fin 8962  df-fi 9429  df-rest 17398  df-topgen 17419  df-top 22790  df-topon 22807  df-bases 22843  df-cld 22917  df-cls 22919

This theorem is referenced by:  restclsseplem  47924