Theorem mrerintcl 17550

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

Assertion

Ref	Expression
mrerintcl	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)

Proof of Theorem mrerintcl

Step	Hyp	Ref	Expression
1		rint0 4931	. . . 4 ⊢ (𝑆 = ∅ → (𝑋 ∩ ∩ 𝑆) = 𝑋)
2	1	adantl 481	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) = 𝑋)
3		mre1cl 17547	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 727	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2837	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
6		simp2 1138	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝐶)
7		mresspw 17545	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
8	7	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
9	6, 8	sstrd 3933	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10		simp3 1139	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11		rintn0 5052	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
12	9, 10, 11	syl2anc 585	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
13		mreintcl 17548	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶)
14	12, 13	eqeltrd 2837	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
15	14	3expa 1119	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
16	5, 15	pm2.61dane 3020	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∩ cint 4890  ‘cfv 6492  Moorecmre 17535

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 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

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 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-mre 17539

This theorem is referenced by:  mreacs  17615  topmtcl  36561