Theorem mrcss 17582

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 3928	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
2	1	adantr 480	. . . . 5 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑠 ∈ 𝐶) → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
3	2	ss2rabdv 4015	. . . 4 ⊢ (𝑈 ⊆ 𝑉 → {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠} ⊆ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
4		intss 4911	. . . 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 3930	. . . 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 17576	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
12	7, 9, 11	syl2anc 585	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
13	10	mrcval 17576	. . 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 3977	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3389   ⊆ wss 3889  ∩ cint 4889  ‘cfv 6498  Moorecmre 17544  mrClscmrc 17545

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-mre 17548  df-mrc 17549

This theorem is referenced by:  mrcsscl  17586  mrcuni  17587  mrcssd  17590  ismrc  43133  isnacs3  43142