MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcval Structured version   Visualization version   GIF version
Theorem mrcval 17550
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 17548 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
32adantr 481 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
4 sseq1 4006 . . . . 5 (π‘₯ = π‘ˆ β†’ (π‘₯ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑠))
54rabbidv 3440 . . . 4 (π‘₯ = π‘ˆ β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
65inteqd 4954 . . 3 (π‘₯ = π‘ˆ β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
76adantl 482 . 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 17534 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
9 elpw2g 5343 . . . 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 478 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
12 sseq2 4007 . . . . 5 (𝑠 = 𝑋 β†’ (π‘ˆ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑋))
138adantr 481 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ 𝐢)
14 simpr 485 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
1512, 13, 14elrabd 3684 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
1615ne0d 4334 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} β‰  βˆ…)
17 intex 5336 . . 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 7002 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  Vcvv 3474   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βˆ© cint 4949   ↦ cmpt 5230  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:  mrcid  17553  mrcss  17556  mrcssid  17557  cycsubg2  19081  aspval2  21443  mrelatlubALT  47573  mreclat  47575
  Copyright terms: Public domain W3C validator