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Theorem mrcfval 16184
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 6175 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3584 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ran Moore)
4 unieq 4415 . . . . . . 7 (𝑐 = 𝐶 𝑐 = 𝐶)
54pweqd 4140 . . . . . 6 (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
6 rabeq 3184 . . . . . . 7 (𝑐 = 𝐶 → {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
76inteqd 4450 . . . . . 6 (𝑐 = 𝐶 {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
85, 7mpteq12dv 4698 . . . . 5 (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
9 df-mrc 16163 . . . . 5 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
10 mreunirn 16177 . . . . . . . 8 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
11 mrcflem 16182 . . . . . . . 8 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
1210, 11sylbi 207 . . . . . . 7 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
13 fssxp 6019 . . . . . . 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 6908 . . . . . . . 8 𝑐 ∈ V
1615pwex 4813 . . . . . . 7 𝒫 𝑐 ∈ V
17 vex 3194 . . . . . . 7 𝑐 ∈ V
1816, 17xpex 6916 . . . . . 6 (𝒫 𝑐 × 𝑐) ∈ V
19 ssexg 4769 . . . . . 6 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
2014, 18, 19sylancl 693 . . . . 5 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
218, 9, 20fvmpt3 6244 . . . 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 16176 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
2423pweqd 4140 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝒫 𝐶 = 𝒫 𝑋)
2524mpteq1d 4703 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
2622, 25eqtrd 2660 . 2 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
271, 26syl5eq 2672 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  {crab 2916  Vcvv 3191  wss 3560  𝒫 cpw 4135   cuni 4407   cint 4445  cmpt 4678   × cxp 5077  ran crn 5080  wf 5846  cfv 5850  Moorecmre 16158  mrClscmrc 16159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-mre 16162  df-mrc 16163
This theorem is referenced by:  mrcf  16185  mrcval  16186  acsficl2d  17092  mrclsp  18903  mrccls  20788
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