Theorem mrerintcl 17640

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 4988	. . . 4 ⊢ (𝑆 = ∅ → (𝑋 ∩ ∩ 𝑆) = 𝑋)
2	1	adantl 481	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) = 𝑋)
3		mre1cl 17637	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 726	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2841	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
6		simp2 1138	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝐶)
7		mresspw 17635	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
8	7	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
9	6, 8	sstrd 3994	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10		simp3 1139	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11		rintn0 5109	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
12	9, 10, 11	syl2anc 584	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
13		mreintcl 17638	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶)
14	12, 13	eqeltrd 2841	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
15	14	3expa 1119	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
16	5, 15	pm2.61dane 3029	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  ∩ cint 4946  ‘cfv 6561  Moorecmre 17625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mre 17629

This theorem is referenced by:  mreacs  17701  topmtcl  36364