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Theorem submrc 17608
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 = (mrClsβ€˜πΆ)
submrc.g 𝐺 = (mrClsβ€˜(𝐢 ∩ 𝒫 𝐷))
Assertion
Ref Expression
submrc ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) = (πΉβ€˜π‘ˆ))
Proof of Theorem submrc
StepHypRef Expression
1 submre 17585 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢) β†’ (𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·))
213adant3 1130 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·))
3 simp1 1134 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
4 submrc.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
5 simp3 1136 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† 𝐷)
6 mress 17573 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢) β†’ 𝐷 βŠ† 𝑋)
763adant3 1130 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ 𝐷 βŠ† 𝑋)
85, 7sstrd 3990 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† 𝑋)
93, 4, 8mrcssidd 17605 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† (πΉβ€˜π‘ˆ))
104mrccl 17591 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) ∈ 𝐢)
113, 8, 10syl2anc 583 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) ∈ 𝐢)
124mrcsscl 17600 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝐷 ∧ 𝐷 ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
13123com23 1124 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
14 fvex 6910 . . . . . 6 (πΉβ€˜π‘ˆ) ∈ V
1514elpw 4607 . . . . 5 ((πΉβ€˜π‘ˆ) ∈ 𝒫 𝐷 ↔ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
1613, 15sylibr 233 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) ∈ 𝒫 𝐷)
1711, 16elind 4194 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) ∈ (𝐢 ∩ 𝒫 𝐷))
18 submrc.g . . . 4 𝐺 = (mrClsβ€˜(𝐢 ∩ 𝒫 𝐷))
1918mrcsscl 17600 . . 3 (((𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·) ∧ π‘ˆ βŠ† (πΉβ€˜π‘ˆ) ∧ (πΉβ€˜π‘ˆ) ∈ (𝐢 ∩ 𝒫 𝐷)) β†’ (πΊβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘ˆ))
202, 9, 17, 19syl3anc 1369 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘ˆ))
212, 18, 5mrcssidd 17605 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† (πΊβ€˜π‘ˆ))
2218mrccl 17591 . . . . 5 (((𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·) ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ (𝐢 ∩ 𝒫 𝐷))
232, 5, 22syl2anc 583 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ (𝐢 ∩ 𝒫 𝐷))
2423elin1d 4198 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ 𝐢)
254mrcsscl 17600 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† (πΊβ€˜π‘ˆ) ∧ (πΊβ€˜π‘ˆ) ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΊβ€˜π‘ˆ))
263, 21, 24, 25syl3anc 1369 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΊβ€˜π‘ˆ))
2720, 26eqssd 3997 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) = (πΉβ€˜π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4603  β€˜cfv 6548  Moorecmre 17562  mrClscmrc 17563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-mre 17566  df-mrc 17567
This theorem is referenced by:  evlseu  22029
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