Theorem submrc 16901

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 16878	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
2	1	3adant3 1128	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3		simp1 1132	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐶 ∈ (Moore‘𝑋))
4		submrc.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
5		simp3 1134	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝐷)
6		mress 16866	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → 𝐷 ⊆ 𝑋)
7	6	3adant3 1128	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐷 ⊆ 𝑋)
8	5, 7	sstrd 3979	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝑋)
9	3, 4, 8	mrcssidd 16898	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐹‘𝑈))
10	4	mrccl 16884	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
11	3, 8, 10	syl2anc 586	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝐶)
12	4	mrcsscl 16893	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐷 ∧ 𝐷 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝐷)
13	12	3com23 1122	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ 𝐷)
14		fvex 6685	. . . . . 6 ⊢ (𝐹‘𝑈) ∈ V
15	14	elpw 4545	. . . . 5 ⊢ ((𝐹‘𝑈) ∈ 𝒫 𝐷 ↔ (𝐹‘𝑈) ⊆ 𝐷)
16	13, 15	sylibr 236	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝒫 𝐷)
17	11, 16	elind 4173	. . 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 16893	. . 3 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹‘𝑈) ∧ (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
20	2, 9, 17, 19	syl3anc 1367	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
21	2, 18, 5	mrcssidd 16898	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐺‘𝑈))
22	18	mrccl 16884	. . . . 5 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
23	2, 5, 22	syl2anc 586	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
24	23	elin1d 4177	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ 𝐶)
25	4	mrcsscl 16893	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺‘𝑈) ∧ (𝐺‘𝑈) ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
26	3, 21, 24, 25	syl3anc 1367	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
27	20, 26	eqssd 3986	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541  ‘cfv 6357  Moorecmre 16855  mrClscmrc 16856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-mre 16859  df-mrc 16860

This theorem is referenced by:  evlseu  20298