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Theorem submrc 17568
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 17545 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢) β†’ (𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·))
213adant3 1132 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·))
3 simp1 1136 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
4 submrc.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
5 simp3 1138 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† 𝐷)
6 mress 17533 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢) β†’ 𝐷 βŠ† 𝑋)
763adant3 1132 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ 𝐷 βŠ† 𝑋)
85, 7sstrd 3991 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† 𝑋)
93, 4, 8mrcssidd 17565 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† (πΉβ€˜π‘ˆ))
104mrccl 17551 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) ∈ 𝐢)
113, 8, 10syl2anc 584 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) ∈ 𝐢)
124mrcsscl 17560 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝐷 ∧ 𝐷 ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
13123com23 1126 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
14 fvex 6901 . . . . . 6 (πΉβ€˜π‘ˆ) ∈ V
1514elpw 4605 . . . . 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 4193 . . 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 17560 . . 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 1371 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘ˆ))
212, 18, 5mrcssidd 17565 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† (πΊβ€˜π‘ˆ))
2218mrccl 17551 . . . . 5 (((𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·) ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ (𝐢 ∩ 𝒫 𝐷))
232, 5, 22syl2anc 584 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ (𝐢 ∩ 𝒫 𝐷))
2423elin1d 4197 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ 𝐢)
254mrcsscl 17560 . . 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 1371 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΊβ€˜π‘ˆ))
2720, 26eqssd 3998 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 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  β€˜cfv 6540  Moorecmre 17522  mrClscmrc 17523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-mre 17526  df-mrc 17527
This theorem is referenced by:  evlseu  21637
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