Theorem submrc 16288

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 16265	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
2	1	3adant3 1081	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3		simp1 1061	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐶 ∈ (Moore‘𝑋))
4		submrc.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
5		simp3 1063	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝐷)
6		mress 16253	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → 𝐷 ⊆ 𝑋)
7	6	3adant3 1081	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐷 ⊆ 𝑋)
8	5, 7	sstrd 3613	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝑋)
9	3, 4, 8	mrcssidd 16285	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐹‘𝑈))
10	4	mrccl 16271	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
11	3, 8, 10	syl2anc 693	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝐶)
12	4	mrcsscl 16280	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐷 ∧ 𝐷 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝐷)
13	12	3com23 1271	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ 𝐷)
14		fvex 6201	. . . . . 6 ⊢ (𝐹‘𝑈) ∈ V
15	14	elpw 4164	. . . . 5 ⊢ ((𝐹‘𝑈) ∈ 𝒫 𝐷 ↔ (𝐹‘𝑈) ⊆ 𝐷)
16	13, 15	sylibr 224	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝒫 𝐷)
17	11, 16	elind 3798	. . 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 16280	. . 3 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹‘𝑈) ∧ (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
20	2, 9, 17, 19	syl3anc 1326	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
21	2, 18, 5	mrcssidd 16285	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐺‘𝑈))
22		inss1 3833	. . . 4 ⊢ (𝐶 ∩ 𝒫 𝐷) ⊆ 𝐶
23	18	mrccl 16271	. . . . 5 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
24	2, 5, 23	syl2anc 693	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
25	22, 24	sseldi 3601	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ 𝐶)
26	4	mrcsscl 16280	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺‘𝑈) ∧ (𝐺‘𝑈) ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
27	3, 21, 25, 26	syl3anc 1326	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
28	20, 27	eqssd 3620	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ‘cfv 5888  Moorecmre 16242  mrClscmrc 16243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by:  evlseu  19516