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Theorem mrcval 17653
Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcval ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
Distinct variable groups:   𝐹,𝑠   𝐶,𝑠   𝑋,𝑠   𝑈,𝑠

Proof of Theorem mrcval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
21mrcfval 17651 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
32adantr 480 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
4 sseq1 4009 . . . . 5 (𝑥 = 𝑈 → (𝑥𝑠𝑈𝑠))
54rabbidv 3444 . . . 4 (𝑥 = 𝑈 → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
65inteqd 4951 . . 3 (𝑥 = 𝑈 {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
76adantl 481 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) ∧ 𝑥 = 𝑈) → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
8 mre1cl 17637 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
9 elpw2g 5333 . . . 4 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
108, 9syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
1110biimpar 477 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
12 sseq2 4010 . . . . 5 (𝑠 = 𝑋 → (𝑈𝑠𝑈𝑋))
138adantr 480 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋𝐶)
14 simpr 484 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈𝑋)
1512, 13, 14elrabd 3694 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋 ∈ {𝑠𝐶𝑈𝑠})
1615ne0d 4342 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ≠ ∅)
17 intex 5344 . . 3 ({𝑠𝐶𝑈𝑠} ≠ ∅ ↔ {𝑠𝐶𝑈𝑠} ∈ V)
1816, 17sylib 218 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ∈ V)
193, 7, 11, 18fvmptd 7023 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  {crab 3436  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600   cint 4946  cmpt 5225  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-sbc 3789  df-csb 3900  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:  mrcid  17656  mrcss  17659  mrcssid  17660  cycsubg2  19228  aspval2  21918  mrelatlubALT  48884  mreclat  48886
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