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Theorem mrcval 17595
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 17593 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
32adantr 479 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
4 sseq1 4005 . . . . 5 (π‘₯ = π‘ˆ β†’ (π‘₯ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑠))
54rabbidv 3436 . . . 4 (π‘₯ = π‘ˆ β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
65inteqd 4956 . . 3 (π‘₯ = π‘ˆ β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
76adantl 480 . 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 17579 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ 𝐢)
9 elpw2g 5348 . . . 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 476 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ π‘ˆ ∈ 𝒫 𝑋)
12 sseq2 4006 . . . . 5 (𝑠 = 𝑋 β†’ (π‘ˆ βŠ† 𝑠 ↔ π‘ˆ βŠ† 𝑋))
138adantr 479 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ 𝑋 ∈ 𝐢)
14 simpr 483 . . . . 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 4337 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} β‰  βˆ…)
17 intex 5341 . . 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 7015 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 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  {crab 3428  Vcvv 3471   βŠ† wss 3947  βˆ…c0 4324  π’« cpw 4604  βˆ© cint 4951   ↦ cmpt 5233  β€˜cfv 6551  Moorecmre 17567  mrClscmrc 17568
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-mre 17571  df-mrc 17572
This theorem is referenced by:  mrcid  17598  mrcss  17601  mrcssid  17602  cycsubg2  19170  aspval2  21836  mrelatlubALT  48057  mreclat  48059
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