Theorem mrerintcl 17645

Description: The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
mrerintcl	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)

Proof of Theorem mrerintcl

Step	Hyp	Ref	Expression
1		rint0 4954	. . . 4 ⊢ (𝑆 = ∅ → (𝑋 ∩ ∩ 𝑆) = 𝑋)
2	1	adantl 486	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) = 𝑋)
3		mre1cl 17642	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 738	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2869	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
6		simp2 1153	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝐶)
7		mresspw 17640	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
8	7	3ad2ant1 1149	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
9	6, 8	sstrd 3955	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10		simp3 1154	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11		rintn0 5076	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
12	9, 10, 11	syl2anc 595	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
13		mreintcl 17643	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶)
14	12, 13	eqeltrd 2869	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
15	14	3expa 1134	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
16	5, 15	pm2.61dane 3051	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  ∩ cint 4913  ‘cfv 6534  Moorecmre 17630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-mre 17634

This theorem is referenced by:  mreacs  17710  topmtcl  36759