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Theorem mrcfval 17651
Description: Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcfval (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Distinct variable groups:   𝑥,𝐹,𝑠   𝑥,𝐶,𝑠   𝑥,𝑋,𝑠

Proof of Theorem mrcfval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . 2 𝐹 = (mrCls‘𝐶)
2 fvssunirn 6939 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3979 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ran Moore)
4 unieq 4918 . . . . . . 7 (𝑐 = 𝐶 𝑐 = 𝐶)
54pweqd 4617 . . . . . 6 (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
6 rabeq 3451 . . . . . . 7 (𝑐 = 𝐶 → {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
76inteqd 4951 . . . . . 6 (𝑐 = 𝐶 {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
85, 7mpteq12dv 5233 . . . . 5 (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
9 df-mrc 17630 . . . . 5 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
10 mreunirn 17644 . . . . . . . 8 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
11 mrcflem 17649 . . . . . . . 8 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
1210, 11sylbi 217 . . . . . . 7 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
13 fssxp 6763 . . . . . . 7 ((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐))
1412, 13syl 17 . . . . . 6 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐))
15 vuniex 7759 . . . . . . . 8 𝑐 ∈ V
1615pwex 5380 . . . . . . 7 𝒫 𝑐 ∈ V
17 vex 3484 . . . . . . 7 𝑐 ∈ V
1816, 17xpex 7773 . . . . . 6 (𝒫 𝑐 × 𝑐) ∈ V
19 ssexg 5323 . . . . . 6 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
2014, 18, 19sylancl 586 . . . . 5 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
218, 9, 20fvmpt3 7020 . . . 4 (𝐶 ran Moore → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
223, 21syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
23 mreuni 17643 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
2423pweqd 4617 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝒫 𝐶 = 𝒫 𝑋)
2524mpteq1d 5237 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
2622, 25eqtrd 2777 . 2 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
271, 26eqtrid 2789 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907   cint 4946  cmpt 5225   × cxp 5683  ran crn 5686  wf 6557  cfv 6561  Moorecmre 17625  mrClscmrc 17626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630
This theorem is referenced by:  mrcf  17652  mrcval  17653  acsficl2d  18597  mrclsp  20987  mrccls  23087
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