Theorem mrcss 17628

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 3965	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
2	1	adantr 480	. . . . 5 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑠 ∈ 𝐶) → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
3	2	ss2rabdv 4051	. . . 4 ⊢ (𝑈 ⊆ 𝑉 → {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠} ⊆ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
4		intss 4945	. . . 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 1134	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ⊆ ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
7		simp1 1136	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
8		sstr 3967	. . . 4 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
9	8	3adant1 1130	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
10		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
11	10	mrcval 17622	. . 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 17622	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑉) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
14	13	3adant2 1131	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑉) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠})
15	6, 12, 14	3sstr4d 4014	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  {crab 3415   ⊆ wss 3926  ∩ cint 4922  ‘cfv 6531  Moorecmre 17594  mrClscmrc 17595

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-mre 17598  df-mrc 17599

This theorem is referenced by:  mrcsscl  17632  mrcuni  17633  mrcssd  17636  ismrc  42724  isnacs3  42733