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Theorem submrc 17014
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 16991 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
213adant3 1133 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3 simp1 1137 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐶 ∈ (Moore‘𝑋))
4 submrc.f . . . 4 𝐹 = (mrCls‘𝐶)
5 simp3 1139 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝐷)
6 mress 16979 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶) → 𝐷𝑋)
763adant3 1133 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝐷𝑋)
85, 7sstrd 3897 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈𝑋)
93, 4, 8mrcssidd 17011 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐹𝑈))
104mrccl 16997 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
113, 8, 10syl2anc 587 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝐶)
124mrcsscl 17006 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐷𝐷𝐶) → (𝐹𝑈) ⊆ 𝐷)
13123com23 1127 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ 𝐷)
14 fvex 6699 . . . . . 6 (𝐹𝑈) ∈ V
1514elpw 4502 . . . . 5 ((𝐹𝑈) ∈ 𝒫 𝐷 ↔ (𝐹𝑈) ⊆ 𝐷)
1613, 15sylibr 237 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ 𝒫 𝐷)
1711, 16elind 4094 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18 submrc.g . . . 4 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
1918mrcsscl 17006 . . 3 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹𝑈) ∧ (𝐹𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺𝑈) ⊆ (𝐹𝑈))
202, 9, 17, 19syl3anc 1372 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ⊆ (𝐹𝑈))
212, 18, 5mrcssidd 17011 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → 𝑈 ⊆ (𝐺𝑈))
2218mrccl 16997 . . . . 5 (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
232, 5, 22syl2anc 587 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
2423elin1d 4098 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) ∈ 𝐶)
254mrcsscl 17006 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺𝑈) ∧ (𝐺𝑈) ∈ 𝐶) → (𝐹𝑈) ⊆ (𝐺𝑈))
263, 21, 24, 25syl3anc 1372 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐹𝑈) ⊆ (𝐺𝑈))
2720, 26eqssd 3904 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  cin 3852  wss 3853  𝒫 cpw 4498  cfv 6349  Moorecmre 16968  mrClscmrc 16969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-int 4847  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-mre 16972  df-mrc 16973
This theorem is referenced by:  evlseu  20909
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