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Theorem submrc 17571
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 17548 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
213adant3 1129 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3 simp1 1133 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐶 ∈ (Moore‘𝑋))
4 submrc.f . . . 4 𝐹 = (mrCls‘𝐶)
5 simp3 1135 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝐷)
6 mress 17536 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → 𝐷𝑋)
763adant3 1129 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐷𝑋)
85, 7sstrd 3984 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝑋)
93, 4, 8mrcssidd 17568 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐹𝑈))
104mrccl 17554 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
113, 8, 10syl2anc 583 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝐶)
124mrcsscl 17563 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐷𝐷𝐶) → (𝐹𝑈) ⊆ 𝐷)
13123com23 1123 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ 𝐷)
14 fvex 6894 . . . . . 6 (𝐹𝑈) ∈ V
1514elpw 4598 . . . . 5 ((𝐹𝑈) ∈ 𝒫 𝐷 ↔ (𝐹𝑈) ⊆ 𝐷)
1613, 15sylibr 233 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝒫 𝐷)
1711, 16elind 4186 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18 submrc.g . . . 4 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
1918mrcsscl 17563 . . 3 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹𝑈) ∧ (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺𝑈) ⊆ (𝐹𝑈))
202, 9, 17, 19syl3anc 1368 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ⊆ (𝐹𝑈))
212, 18, 5mrcssidd 17568 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐺𝑈))
2218mrccl 17554 . . . . 5 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
232, 5, 22syl2anc 583 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
2423elin1d 4190 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ 𝐶)
254mrcsscl 17563 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺𝑈) ∧ (𝐺𝑈) ∈ 𝐶) → (𝐹𝑈) ⊆ (𝐺𝑈))
263, 21, 24, 25syl3anc 1368 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ (𝐺𝑈))
2720, 26eqssd 3991 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cin 3939  wss 3940  𝒫 cpw 4594  cfv 6533  Moorecmre 17525  mrClscmrc 17526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-mre 17529  df-mrc 17530
This theorem is referenced by:  evlseu  21956
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