Theorem mrcss 17674

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 4015	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
2	1	adantr 480	. . . . 5 ⊢ ((𝑈 ⊆ 𝑉 ∧ 𝑠 ∈ 𝐶) → (𝑉 ⊆ 𝑠 → 𝑈 ⊆ 𝑠))
3	2	ss2rabdv 4099	. . . 4 ⊢ (𝑈 ⊆ 𝑉 → {𝑠 ∈ 𝐶 ∣ 𝑉 ⊆ 𝑠} ⊆ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
4		intss 4993	. . . 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 4017	. . . 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 17668	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
12	7, 9, 11	syl2anc 583	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
13	10	mrcval 17668	. . 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 4056	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443   ⊆ wss 3976  ∩ cint 4970  ‘cfv 6573  Moorecmre 17640  mrClscmrc 17641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-mre 17644  df-mrc 17645

This theorem is referenced by:  mrcsscl  17678  mrcuni  17679  mrcssd  17682  ismrc  42657  isnacs3  42666