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Theorem mrcfval 17111
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 6746 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3896 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ran Moore)
4 unieq 4830 . . . . . . 7 (𝑐 = 𝐶 𝑐 = 𝐶)
54pweqd 4532 . . . . . 6 (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
6 rabeq 3394 . . . . . . 7 (𝑐 = 𝐶 → {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
76inteqd 4864 . . . . . 6 (𝑐 = 𝐶 {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
85, 7mpteq12dv 5140 . . . . 5 (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
9 df-mrc 17090 . . . . 5 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
10 mreunirn 17104 . . . . . . . 8 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
11 mrcflem 17109 . . . . . . . 8 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
1210, 11sylbi 220 . . . . . . 7 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
13 fssxp 6573 . . . . . . 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 7527 . . . . . . . 8 𝑐 ∈ V
1615pwex 5273 . . . . . . 7 𝒫 𝑐 ∈ V
17 vex 3412 . . . . . . 7 𝑐 ∈ V
1816, 17xpex 7538 . . . . . 6 (𝒫 𝑐 × 𝑐) ∈ V
19 ssexg 5216 . . . . . 6 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
2014, 18, 19sylancl 589 . . . . 5 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
218, 9, 20fvmpt3 6822 . . . 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 17103 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
2423pweqd 4532 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝒫 𝐶 = 𝒫 𝑋)
2524mpteq1d 5144 . . 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, 26syl5eq 2790 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  {crab 3065  Vcvv 3408  wss 3866  𝒫 cpw 4513   cuni 4819   cint 4859  cmpt 5135   × cxp 5549  ran crn 5552  wf 6376  cfv 6380  Moorecmre 17085  mrClscmrc 17086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-mre 17089  df-mrc 17090
This theorem is referenced by:  mrcf  17112  mrcval  17113  acsficl2d  18058  mrclsp  20026  mrccls  21976
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