Theorem restcls2 48028

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 6906	. . 3 ⊢ (𝜑 → (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌)))
3	2	fveq1d 6904	. 2 ⊢ (𝜑 → ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆))
4		restcls2.5	. . 3 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
5		cldcls 22974	. . 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 4022	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽)
11	7, 9, 8, 1, 4	restcls2lem 48027	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12		eqid 2728	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
13		eqid 2728	. . . 4 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌)
14	12, 13	restcls 23113	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
15	7, 10, 11, 14	syl3anc 1368	. 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 1533   ∈ wcel 2098   ∩ cin 3948   ⊆ wss 3949  ∪ cuni 4912  ‘cfv 6553  (class class class)co 7426   ↾_t crest 17411  Topctop 22823  Clsdccld 22948  clsccl 22950

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-en 8973  df-fin 8976  df-fi 9444  df-rest 17413  df-topgen 17434  df-top 22824  df-topon 22841  df-bases 22877  df-cld 22951  df-cls 22953

This theorem is referenced by:  restclsseplem  48029