Theorem mreriincl 17522

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 5038	. . . 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 17518	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 727	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2837	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 = ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
6		mress 17517	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
7	6	ex 412	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
8	7	ralimdv 3151	. . . . 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 5039	. . . 4 ⊢ ((∀𝑦 ∈ 𝐼 𝑆 ⊆ 𝑋 ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
11	9, 10	sylan 581	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) = ∩ 𝑦 ∈ 𝐼 𝑆)
12		simpll 767	. . . 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 769	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
15		mreiincl 17520	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
16	12, 13, 14, 15	syl3anc 1374	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
17	11, 16	eqeltrd 2837	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
18	5, 17	pm2.61dane 3020	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  ∩ ciin 4948  ‘cfv 6493  Moorecmre 17506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-mre 17510

This theorem is referenced by:  acsfn1  17589  acsfn1c  17590  acsfn2  17591  acsfn1p  20737