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Theorem mrcval 16623
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 16621 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
32adantr 474 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
4 sseq1 3851 . . . . 5 (𝑥 = 𝑈 → (𝑥𝑠𝑈𝑠))
54rabbidv 3402 . . . 4 (𝑥 = 𝑈 → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
65inteqd 4702 . . 3 (𝑥 = 𝑈 {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
76adantl 475 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) ∧ 𝑥 = 𝑈) → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
8 mre1cl 16607 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
9 elpw2g 5049 . . . 4 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
108, 9syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
1110biimpar 471 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
128adantr 474 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋𝐶)
13 simpr 479 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈𝑋)
14 sseq2 3852 . . . . . 6 (𝑠 = 𝑋 → (𝑈𝑠𝑈𝑋))
1514elrab 3585 . . . . 5 (𝑋 ∈ {𝑠𝐶𝑈𝑠} ↔ (𝑋𝐶𝑈𝑋))
1612, 13, 15sylanbrc 580 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋 ∈ {𝑠𝐶𝑈𝑠})
1716ne0d 4151 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ≠ ∅)
18 intex 5042 . . 3 ({𝑠𝐶𝑈𝑠} ≠ ∅ ↔ {𝑠𝐶𝑈𝑠} ∈ V)
1917, 18sylib 210 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ∈ V)
203, 7, 11, 19fvmptd 6535 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wne 2999  {crab 3121  Vcvv 3414  wss 3798  c0 4144  𝒫 cpw 4378   cint 4697  cmpt 4952  cfv 6123  Moorecmre 16595  mrClscmrc 16596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-mre 16599  df-mrc 16600
This theorem is referenced by:  mrcid  16626  mrcss  16629  mrcssid  16630  cycsubg2  17982  aspval2  19708
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