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Theorem mrcval 17497
Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcval ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
Distinct variable groups:   𝐹,𝑠   𝐢,𝑠   𝑋,𝑠   π‘ˆ,𝑠

Proof of Theorem mrcval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
21mrcfval 17495 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
32adantr 482 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
4 sseq1 3974 . . . . 5 (π‘₯ = π‘ˆ β†’ (π‘₯ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑠))
54rabbidv 3418 . . . 4 (π‘₯ = π‘ˆ β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
65inteqd 4917 . . 3 (π‘₯ = π‘ˆ β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
76adantl 483 . 2 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) ∧ π‘₯ = π‘ˆ) β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
8 mre1cl 17481 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
9 elpw2g 5306 . . . 4 (𝑋 ∈ 𝐢 β†’ (π‘ˆ ∈ 𝒫 𝑋 ↔ π‘ˆ βŠ† 𝑋))
108, 9syl 17 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (π‘ˆ ∈ 𝒫 𝑋 ↔ π‘ˆ βŠ† 𝑋))
1110biimpar 479 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
12 sseq2 3975 . . . . 5 (𝑠 = 𝑋 β†’ (π‘ˆ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑋))
138adantr 482 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ 𝐢)
14 simpr 486 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
1512, 13, 14elrabd 3652 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
1615ne0d 4300 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} β‰  βˆ…)
17 intex 5299 . . 3 ({𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} β‰  βˆ… ↔ ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} ∈ V)
1816, 17sylib 217 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} ∈ V)
193, 7, 11, 18fvmptd 6960 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆ© cint 4912   ↦ cmpt 5193  β€˜cfv 6501  Moorecmre 17469  mrClscmrc 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-mre 17473  df-mrc 17474
This theorem is referenced by:  mrcid  17500  mrcss  17503  mrcssid  17504  cycsubg2  19010  aspval2  21317  mrelatlubALT  47094  mreclat  47096
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