Theorem submrc 17673

Description: In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
submrc.f	⊢ 𝐹 = (mrCls‘𝐶)
submrc.g	⊢ 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))

Assertion

Ref	Expression
submrc	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

Proof of Theorem submrc

Step	Hyp	Ref	Expression
1		submre 17650	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
2	1	3adant3 1131	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3		simp1 1135	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐶 ∈ (Moore‘𝑋))
4		submrc.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
5		simp3 1137	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝐷)
6		mress 17638	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → 𝐷 ⊆ 𝑋)
7	6	3adant3 1131	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐷 ⊆ 𝑋)
8	5, 7	sstrd 4006	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝑋)
9	3, 4, 8	mrcssidd 17670	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐹‘𝑈))
10	4	mrccl 17656	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
11	3, 8, 10	syl2anc 584	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝐶)
12	4	mrcsscl 17665	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐷 ∧ 𝐷 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝐷)
13	12	3com23 1125	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ 𝐷)
14		fvex 6920	. . . . . 6 ⊢ (𝐹‘𝑈) ∈ V
15	14	elpw 4609	. . . . 5 ⊢ ((𝐹‘𝑈) ∈ 𝒫 𝐷 ↔ (𝐹‘𝑈) ⊆ 𝐷)
16	13, 15	sylibr 234	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝒫 𝐷)
17	11, 16	elind 4210	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18		submrc.g	. . . 4 ⊢ 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
19	18	mrcsscl 17665	. . 3 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹‘𝑈) ∧ (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
20	2, 9, 17, 19	syl3anc 1370	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
21	2, 18, 5	mrcssidd 17670	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐺‘𝑈))
22	18	mrccl 17656	. . . . 5 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
23	2, 5, 22	syl2anc 584	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
24	23	elin1d 4214	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ 𝐶)
25	4	mrcsscl 17665	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺‘𝑈) ∧ (𝐺‘𝑈) ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
26	3, 21, 24, 25	syl3anc 1370	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
27	20, 26	eqssd 4013	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ‘cfv 6563  Moorecmre 17627  mrClscmrc 17628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mre 17631  df-mrc 17632

This theorem is referenced by:  evlseu  22125