Theorem mrerintcl 17616

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 4943	. . . 4 ⊢ (𝑆 = ∅ → (𝑋 ∩ ∩ 𝑆) = 𝑋)
2	1	adantl 485	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) = 𝑋)
3		mre1cl 17613	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
4	3	ad2antrr 736	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → 𝑋 ∈ 𝐶)
5	2, 4	eqeltrd 2861	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
6		simp2 1149	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝐶)
7		mresspw 17611	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
8	7	3ad2ant1 1145	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
9	6, 8	sstrd 3944	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10		simp3 1150	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11		rintn0 5063	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
12	9, 10, 11	syl2anc 593	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆)
13		mreintcl 17614	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶)
14	12, 13	eqeltrd 2861	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
15	14	3expa 1130	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
16	5, 15	pm2.61dane 3043	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552  ∩ cint 4902  ‘cfv 6516  Moorecmre 17601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-mre 17605

This theorem is referenced by:  mreacs  17681  topmtcl  36684