Theorem mrcss 17659

Description: Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
mrcss	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))

Proof of Theorem mrcss

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3990	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
2	1	adantr 480	. . . . 5 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑠 ∈ 𝐶) → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
3	2	ss2rabdv 4076	. . . 4 ⊢ (𝑈 ⊆ 𝑉 → {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠} ⊆ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
4		intss 4969	. . . 4 ⊢ ({𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠} ⊆ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ⊆ ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
5	3, 4	syl 17	. . 3 ⊢ (𝑈 ⊆ 𝑉 → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ⊆ ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
6	5	3ad2ant2 1135	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ⊆ ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
7		simp1 1137	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
8		sstr 3992	. . . 4 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
9	8	3adant1 1131	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
10		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
11	10	mrcval 17653	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
12	7, 9, 11	syl2anc 584	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
13	10	mrcval 17653	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑉) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
14	13	3adant2 1132	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑉) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
15	6, 12, 14	3sstr4d 4039	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {crab 3436   ⊆ wss 3951  ∩ cint 4946  ‘cfv 6561  Moorecmre 17625  mrClscmrc 17626

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630

This theorem is referenced by:  mrcsscl  17663  mrcuni  17664  mrcssd  17667  ismrc  42712  isnacs3  42721