Theorem mrcss 16879

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 3922	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
2	1	adantr 484	. . . . 5 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑠 ∈ 𝐶) → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
3	2	ss2rabdv 4003	. . . 4 ⊢ (𝑈 ⊆ 𝑉 → {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠} ⊆ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
4		intss 4859	. . . 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 1131	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ⊆ ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
7		simp1 1133	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
8		sstr 3923	. . . 4 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
9	8	3adant1 1127	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
10		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
11	10	mrcval 16873	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
12	7, 9, 11	syl2anc 587	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
13	10	mrcval 16873	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑉) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
14	13	3adant2 1128	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑉) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
15	6, 12, 14	3sstr4d 3962	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3110   ⊆ wss 3881  ∩ cint 4838  ‘cfv 6324  Moorecmre 16845  mrClscmrc 16846

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-mre 16849  df-mrc 16850

This theorem is referenced by:  mrcsscl  16883  mrcuni  16884  mrcssd  16887  ismrc  39642  isnacs3  39651