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Theorem mrcfval 17423
Description: Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcfval (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
Distinct variable groups:   π‘₯,𝐹,𝑠   π‘₯,𝐢,𝑠   π‘₯,𝑋,𝑠

Proof of Theorem mrcfval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . 2 𝐹 = (mrClsβ€˜πΆ)
2 fvssunirn 6871 . . . . 5 (Mooreβ€˜π‘‹) βŠ† βˆͺ ran Moore
32sseli 3939 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐢 ∈ βˆͺ ran Moore)
4 unieq 4875 . . . . . . 7 (𝑐 = 𝐢 β†’ βˆͺ 𝑐 = βˆͺ 𝐢)
54pweqd 4576 . . . . . 6 (𝑐 = 𝐢 β†’ 𝒫 βˆͺ 𝑐 = 𝒫 βˆͺ 𝐢)
6 rabeq 3420 . . . . . . 7 (𝑐 = 𝐢 β†’ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠})
76inteqd 4911 . . . . . 6 (𝑐 = 𝐢 β†’ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠})
85, 7mpteq12dv 5195 . . . . 5 (𝑐 = 𝐢 β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) = (π‘₯ ∈ 𝒫 βˆͺ 𝐢 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
9 df-mrc 17402 . . . . 5 mrCls = (𝑐 ∈ βˆͺ ran Moore ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}))
10 mreunirn 17416 . . . . . . . 8 (𝑐 ∈ βˆͺ ran Moore ↔ 𝑐 ∈ (Mooreβ€˜βˆͺ 𝑐))
11 mrcflem 17421 . . . . . . . 8 (𝑐 ∈ (Mooreβ€˜βˆͺ 𝑐) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}):𝒫 βˆͺ π‘βŸΆπ‘)
1210, 11sylbi 216 . . . . . . 7 (𝑐 ∈ βˆͺ ran Moore β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}):𝒫 βˆͺ π‘βŸΆπ‘)
13 fssxp 6692 . . . . . . 7 ((π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}):𝒫 βˆͺ π‘βŸΆπ‘ β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) βŠ† (𝒫 βˆͺ 𝑐 Γ— 𝑐))
1412, 13syl 17 . . . . . 6 (𝑐 ∈ βˆͺ ran Moore β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) βŠ† (𝒫 βˆͺ 𝑐 Γ— 𝑐))
15 vuniex 7667 . . . . . . . 8 βˆͺ 𝑐 ∈ V
1615pwex 5334 . . . . . . 7 𝒫 βˆͺ 𝑐 ∈ V
17 vex 3448 . . . . . . 7 𝑐 ∈ V
1816, 17xpex 7678 . . . . . 6 (𝒫 βˆͺ 𝑐 Γ— 𝑐) ∈ V
19 ssexg 5279 . . . . . 6 (((π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) βŠ† (𝒫 βˆͺ 𝑐 Γ— 𝑐) ∧ (𝒫 βˆͺ 𝑐 Γ— 𝑐) ∈ V) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) ∈ V)
2014, 18, 19sylancl 587 . . . . 5 (𝑐 ∈ βˆͺ ran Moore β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) ∈ V)
218, 9, 20fvmpt3 6948 . . . 4 (𝐢 ∈ βˆͺ ran Moore β†’ (mrClsβ€˜πΆ) = (π‘₯ ∈ 𝒫 βˆͺ 𝐢 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
223, 21syl 17 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (mrClsβ€˜πΆ) = (π‘₯ ∈ 𝒫 βˆͺ 𝐢 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
23 mreuni 17415 . . . . 5 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ βˆͺ 𝐢 = 𝑋)
2423pweqd 4576 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝒫 βˆͺ 𝐢 = 𝒫 𝑋)
2524mpteq1d 5199 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝐢 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}) = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
2622, 25eqtrd 2778 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (mrClsβ€˜πΆ) = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
271, 26eqtrid 2790 1 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹 = (π‘₯ ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐢 ∣ π‘₯ βŠ† 𝑠}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444   βŠ† wss 3909  π’« cpw 4559  βˆͺ cuni 4864  βˆ© cint 4906   ↦ cmpt 5187   Γ— cxp 5629  ran crn 5632  βŸΆwf 6488  β€˜cfv 6492  Moorecmre 17397  mrClscmrc 17398
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mre 17401  df-mrc 17402
This theorem is referenced by:  mrcf  17424  mrcval  17425  acsficl2d  18376  mrclsp  20374  mrccls  22353
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