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Theorem mrcval 17561
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 17559 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
32adantr 480 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
4 sseq1 4002 . . . . 5 (π‘₯ = π‘ˆ β†’ (π‘₯ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑠))
54rabbidv 3434 . . . 4 (π‘₯ = π‘ˆ β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
65inteqd 4948 . . 3 (π‘₯ = π‘ˆ β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
76adantl 481 . 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 17545 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
9 elpw2g 5337 . . . 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 477 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
12 sseq2 4003 . . . . 5 (𝑠 = 𝑋 β†’ (π‘ˆ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑋))
138adantr 480 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ 𝐢)
14 simpr 484 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
1512, 13, 14elrabd 3680 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
1615ne0d 4330 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} β‰  βˆ…)
17 intex 5330 . . 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 6998 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426  Vcvv 3468   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  βˆ© cint 4943   ↦ cmpt 5224  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:  mrcid  17564  mrcss  17567  mrcssid  17568  cycsubg2  19134  aspval2  21788  mrelatlubALT  47875  mreclat  47877
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