Theorem mreriincl 17640

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 5044	. . . 4 ⊢ (𝐼 = ∅ → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = 𝑋)
2	1	adantl 486	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = 𝑋)
3		mre1cl 17636	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 738	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2865	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
6		mress 17635	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
7	6	ex 417	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
8	7	ralimdv 3179	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋))
9	8	imp 411	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋)
10		riinn0 5045	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋 ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
11	9, 10	sylan 591	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
12		simpll 778	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13		simpr 489	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14		simplr 780	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
15		mreiincl 17638	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
16	12, 13, 14, 15	syl3anc 1394	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
17	11, 16	eqeltrd 2865	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
18	5, 17	pm2.61dane 3047	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  ∩ ciin 4953  ‘cfv 6525  Moorecmre 17624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-mre 17628

This theorem is referenced by:  acsfn1  17707  acsfn1c  17708  acsfn2  17709  acsfn1p  20871