MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  submrc Structured version   Visualization version   GIF version

Theorem submrc 16603
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) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))

Proof of Theorem submrc
StepHypRef Expression
1 submre 16580 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
213adant3 1163 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3 simp1 1167 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐶 ∈ (Moore‘𝑋))
4 submrc.f . . . 4 𝐹 = (mrCls‘𝐶)
5 simp3 1169 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝐷)
6 mress 16568 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → 𝐷𝑋)
763adant3 1163 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐷𝑋)
85, 7sstrd 3808 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝑋)
93, 4, 8mrcssidd 16600 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐹𝑈))
104mrccl 16586 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
113, 8, 10syl2anc 580 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝐶)
124mrcsscl 16595 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐷𝐷𝐶) → (𝐹𝑈) ⊆ 𝐷)
13123com23 1157 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ 𝐷)
14 fvex 6424 . . . . . 6 (𝐹𝑈) ∈ V
1514elpw 4355 . . . . 5 ((𝐹𝑈) ∈ 𝒫 𝐷 ↔ (𝐹𝑈) ⊆ 𝐷)
1613, 15sylibr 226 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝒫 𝐷)
1711, 16elind 3996 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18 submrc.g . . . 4 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
1918mrcsscl 16595 . . 3 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹𝑈) ∧ (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺𝑈) ⊆ (𝐹𝑈))
202, 9, 17, 19syl3anc 1491 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ⊆ (𝐹𝑈))
212, 18, 5mrcssidd 16600 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐺𝑈))
22 inss1 4028 . . . 4 (𝐶 ∩ 𝒫 𝐷) ⊆ 𝐶
2318mrccl 16586 . . . . 5 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
242, 5, 23syl2anc 580 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
2522, 24sseldi 3796 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ 𝐶)
264mrcsscl 16595 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺𝑈) ∧ (𝐺𝑈) ∈ 𝐶) → (𝐹𝑈) ⊆ (𝐺𝑈))
273, 21, 25, 26syl3anc 1491 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ (𝐺𝑈))
2820, 27eqssd 3815 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108   = wceq 1653  wcel 2157  cin 3768  wss 3769  𝒫 cpw 4349  cfv 6101  Moorecmre 16557  mrClscmrc 16558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-mre 16561  df-mrc 16562
This theorem is referenced by:  evlseu  19838
  Copyright terms: Public domain W3C validator