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Theorem submrc 16891
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 16868 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
213adant3 1126 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3 simp1 1130 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐶 ∈ (Moore‘𝑋))
4 submrc.f . . . 4 𝐹 = (mrCls‘𝐶)
5 simp3 1132 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝐷)
6 mress 16856 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → 𝐷𝑋)
763adant3 1126 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐷𝑋)
85, 7sstrd 3980 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝑋)
93, 4, 8mrcssidd 16888 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐹𝑈))
104mrccl 16874 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
113, 8, 10syl2anc 584 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝐶)
124mrcsscl 16883 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐷𝐷𝐶) → (𝐹𝑈) ⊆ 𝐷)
13123com23 1120 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ 𝐷)
14 fvex 6679 . . . . . 6 (𝐹𝑈) ∈ V
1514elpw 4548 . . . . 5 ((𝐹𝑈) ∈ 𝒫 𝐷 ↔ (𝐹𝑈) ⊆ 𝐷)
1613, 15sylibr 235 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝒫 𝐷)
1711, 16elind 4174 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18 submrc.g . . . 4 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
1918mrcsscl 16883 . . 3 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹𝑈) ∧ (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺𝑈) ⊆ (𝐹𝑈))
202, 9, 17, 19syl3anc 1365 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ⊆ (𝐹𝑈))
212, 18, 5mrcssidd 16888 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐺𝑈))
2218mrccl 16874 . . . . 5 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
232, 5, 22syl2anc 584 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
2423elin1d 4178 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ 𝐶)
254mrcsscl 16883 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺𝑈) ∧ (𝐺𝑈) ∈ 𝐶) → (𝐹𝑈) ⊆ (𝐺𝑈))
263, 21, 24, 25syl3anc 1365 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ (𝐺𝑈))
2720, 26eqssd 3987 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2106  cin 3938  wss 3939  𝒫 cpw 4541  cfv 6351  Moorecmre 16845  mrClscmrc 16846
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-mre 16849  df-mrc 16850
This theorem is referenced by:  evlseu  20216
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