Theorem mreriincl 17566

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 5049	. . . 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 17562	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 726	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2829	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
6		mress 17561	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
7	6	ex 412	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
8	7	ralimdv 3148	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋))
9	8	imp 406	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋)
10		riinn0 5050	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋 ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
11	9, 10	sylan 580	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
12		simpll 766	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13		simpr 484	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14		simplr 768	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
15		mreiincl 17564	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
16	12, 13, 14, 15	syl3anc 1373	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
17	11, 16	eqeltrd 2829	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
18	5, 17	pm2.61dane 3013	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  ∩ ciin 4959  ‘cfv 6514  Moorecmre 17550

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-mre 17554

This theorem is referenced by:  acsfn1  17629  acsfn1c  17630  acsfn2  17631  acsfn1p  20715