Theorem mreriincl 17730

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 5041	. . . 4 ⊢ (𝐼 = ∅ → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = 𝑋)
2	1	adantl 487	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = 𝑋)
3		mre1cl 17726	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 739	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2860	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
6		mress 17725	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
7	6	ex 418	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
8	7	ralimdv 3176	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶 → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋))
9	8	imp 412	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋)
10		riinn0 5042	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋 ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
11	9, 10	sylan 592	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
12		simpll 779	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13		simpr 490	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14		simplr 781	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
15		mreiincl 17728	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
16	12, 13, 14, 15	syl3anc 1398	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
17	11, 16	eqeltrd 2860	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
18	5, 17	pm2.61dane 3042	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  ∩ ciin 4951  ‘cfv 6527  Moorecmre 17714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6483  df-fun 6529  df-fv 6535  df-mre 17718

This theorem is used by:  acsfn1  17797  acsfn1c  17798  acsfn2  17799  acsfn1p  21018