Theorem submrc 17834

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 17807	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
2	1	3adant3 1150	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3		simp1 1154	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐶 ∈ (Moore‘𝑋))
4		submrc.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
5		simp3 1156	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝐷)
6		mress 17795	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → 𝐷 ⊆ 𝑋)
7	6	3adant3 1150	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐷 ⊆ 𝑋)
8	5, 7	sstrd 3923	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝑋)
9	3, 4, 8	mrcssidd 17831	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐹‘𝑈))
10	4	mrccl 17817	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
11	3, 8, 10	syl2anc 596	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝐶)
12	4	mrcsscl 17826	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐷 ∧ 𝐷 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝐷)
13	12	3com23 1144	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ 𝐷)
14		fvex 6924	. . . . . 6 ⊢ (𝐹‘𝑈) ∈ V
15	14	elpw 4564	. . . . 5 ⊢ ((𝐹‘𝑈) ∈ 𝒫 𝐷 ↔ (𝐹‘𝑈) ⊆ 𝐷)
16	13, 15	sylibr 237	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝒫 𝐷)
17	11, 16	elind 4100	. . 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 17826	. . 3 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹‘𝑈) ∧ (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
20	2, 9, 17, 19	syl3anc 1399	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
21	2, 18, 5	mrcssidd 17831	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐺‘𝑈))
22	18	mrccl 17817	. . . . 5 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
23	2, 5, 22	syl2anc 596	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
24	23	elin1d 4104	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ 𝐶)
25	4	mrcsscl 17826	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺‘𝑈) ∧ (𝐺‘𝑈) ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
26	3, 21, 24, 25	syl3anc 1399	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
27	20, 26	eqssd 3930	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ‘cfv 6561  Moorecmre 17784  mrClscmrc 17785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17788  df-mrc 17789

This theorem is used by:  evlseu  22425