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Theorem mrcfval 17652
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 3990 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ran Moore)
4 unieq 4922 . . . . . . 7 (𝑐 = 𝐶 𝑐 = 𝐶)
54pweqd 4621 . . . . . 6 (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
6 rabeq 3447 . . . . . . 7 (𝑐 = 𝐶 → {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
76inteqd 4955 . . . . . 6 (𝑐 = 𝐶 {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
85, 7mpteq12dv 5238 . . . . 5 (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
9 df-mrc 17631 . . . . 5 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
10 mreunirn 17645 . . . . . . . 8 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
11 mrcflem 17650 . . . . . . . 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 7757 . . . . . . . 8 𝑐 ∈ V
1615pwex 5385 . . . . . . 7 𝒫 𝑐 ∈ V
17 vex 3481 . . . . . . 7 𝑐 ∈ V
1816, 17xpex 7771 . . . . . 6 (𝒫 𝑐 × 𝑐) ∈ V
19 ssexg 5328 . . . . . 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 7019 . . . 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 17644 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
2423pweqd 4621 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝒫 𝐶 = 𝒫 𝑋)
2524mpteq1d 5242 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
2622, 25eqtrd 2774 . 2 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
271, 26eqtrid 2786 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  wss 3962  𝒫 cpw 4604   cuni 4911   cint 4950  cmpt 5230   × cxp 5686  ran crn 5689  wf 6558  cfv 6562  Moorecmre 17626  mrClscmrc 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mre 17630  df-mrc 17631
This theorem is referenced by:  mrcf  17653  mrcval  17654  acsficl2d  18609  mrclsp  21004  mrccls  23102
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