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Theorem submrc 17579
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 17556 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢) β†’ (𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·))
213adant3 1129 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (𝐢 ∩ 𝒫 𝐷) ∈ (Mooreβ€˜π·))
3 simp1 1133 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
4 submrc.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
5 simp3 1135 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† 𝐷)
6 mress 17544 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢) β†’ 𝐷 βŠ† 𝑋)
763adant3 1129 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ 𝐷 βŠ† 𝑋)
85, 7sstrd 3987 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† 𝑋)
93, 4, 8mrcssidd 17576 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† (πΉβ€˜π‘ˆ))
104mrccl 17562 . . . . 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 17571 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝐷 ∧ 𝐷 ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
13123com23 1123 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝐷)
14 fvex 6897 . . . . . 6 (πΉβ€˜π‘ˆ) ∈ V
1514elpw 4601 . . . . 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 4189 . . 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 17571 . . 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 1368 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘ˆ))
212, 18, 5mrcssidd 17576 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ π‘ˆ βŠ† (πΊβ€˜π‘ˆ))
2218mrccl 17562 . . . . 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 4193 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΊβ€˜π‘ˆ) ∈ 𝐢)
254mrcsscl 17571 . . 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 1368 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝐷 ∈ 𝐢 ∧ π‘ˆ βŠ† 𝐷) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΊβ€˜π‘ˆ))
2720, 26eqssd 3994 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 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  β€˜cfv 6536  Moorecmre 17533  mrClscmrc 17534
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-mre 17537  df-mrc 17538
This theorem is referenced by:  evlseu  21984
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