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Theorem submrc 16204
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 16181 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
213adant3 1079 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3 simp1 1059 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐶 ∈ (Moore‘𝑋))
4 submrc.f . . . 4 𝐹 = (mrCls‘𝐶)
5 simp3 1061 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝐷)
6 mress 16169 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → 𝐷𝑋)
763adant3 1079 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐷𝑋)
85, 7sstrd 3598 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝑋)
93, 4, 8mrcssidd 16201 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐹𝑈))
104mrccl 16187 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
113, 8, 10syl2anc 692 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝐶)
124mrcsscl 16196 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐷𝐷𝐶) → (𝐹𝑈) ⊆ 𝐷)
13123com23 1268 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ 𝐷)
14 fvex 6160 . . . . . 6 (𝐹𝑈) ∈ V
1514elpw 4141 . . . . 5 ((𝐹𝑈) ∈ 𝒫 𝐷 ↔ (𝐹𝑈) ⊆ 𝐷)
1613, 15sylibr 224 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝒫 𝐷)
1711, 16elind 3781 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18 submrc.g . . . 4 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
1918mrcsscl 16196 . . 3 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹𝑈) ∧ (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺𝑈) ⊆ (𝐹𝑈))
202, 9, 17, 19syl3anc 1323 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ⊆ (𝐹𝑈))
212, 18, 5mrcssidd 16201 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐺𝑈))
22 inss1 3816 . . . 4 (𝐶 ∩ 𝒫 𝐷) ⊆ 𝐶
2318mrccl 16187 . . . . 5 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
242, 5, 23syl2anc 692 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
2522, 24sseldi 3586 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ 𝐶)
264mrcsscl 16196 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺𝑈) ∧ (𝐺𝑈) ∈ 𝐶) → (𝐹𝑈) ⊆ (𝐺𝑈))
273, 21, 25, 26syl3anc 1323 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ (𝐺𝑈))
2820, 27eqssd 3605 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1992  cin 3559  wss 3560  𝒫 cpw 4135  cfv 5850  Moorecmre 16158  mrClscmrc 16159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-mre 16162  df-mrc 16163
This theorem is referenced by:  evlseu  19430
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