Theorem mreriincl 16081

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

Assertion

Ref	Expression
mreriincl	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)

Distinct variable groups:   𝑦,𝐼   𝑦,𝑋   𝑦,𝐶

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem mreriincl

Step	Hyp	Ref	Expression
1		riin0 4530	. . . 4 ⊢ (𝐼 = ∅ → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = 𝑋)
2	1	adantl 481	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = 𝑋)
3		mre1cl 16077	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 758	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2688	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
6		mress 16076	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
7	6	ex 449	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
8	7	ralimdv 2946	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋))
9	8	imp 444	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋)
10		riinn0 4531	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋 ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
11	9, 10	sylan 487	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
12		simpll 786	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13		simpr 476	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14		simplr 788	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
15		mreiincl 16079	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
16	12, 13, 14, 15	syl3anc 1318	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
17	11, 16	eqeltrd 2688	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
18	5, 17	pm2.61dane 2869	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456  ‘cfv 5804  Moorecmre 16065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-mre 16069

This theorem is referenced by:  acsfn1  16145  acsfn1c  16146  acsfn2  16147  acsfn1p  36788