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Theorem mrcval 16492
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 16490 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
32adantr 472 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
4 sseq1 3767 . . . . 5 (𝑥 = 𝑈 → (𝑥𝑠𝑈𝑠))
54rabbidv 3329 . . . 4 (𝑥 = 𝑈 → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
65inteqd 4632 . . 3 (𝑥 = 𝑈 {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
76adantl 473 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) ∧ 𝑥 = 𝑈) → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
8 mre1cl 16476 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
9 elpw2g 4976 . . . 4 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
108, 9syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
1110biimpar 503 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
128adantr 472 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋𝐶)
13 simpr 479 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈𝑋)
14 sseq2 3768 . . . . . 6 (𝑠 = 𝑋 → (𝑈𝑠𝑈𝑋))
1514elrab 3504 . . . . 5 (𝑋 ∈ {𝑠𝐶𝑈𝑠} ↔ (𝑋𝐶𝑈𝑋))
1612, 13, 15sylanbrc 701 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋 ∈ {𝑠𝐶𝑈𝑠})
17 ne0i 4064 . . . 4 (𝑋 ∈ {𝑠𝐶𝑈𝑠} → {𝑠𝐶𝑈𝑠} ≠ ∅)
1816, 17syl 17 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ≠ ∅)
19 intex 4969 . . 3 ({𝑠𝐶𝑈𝑠} ≠ ∅ ↔ {𝑠𝐶𝑈𝑠} ∈ V)
2018, 19sylib 208 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ∈ V)
213, 7, 11, 20fvmptd 6451 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  {crab 3054  Vcvv 3340  wss 3715  c0 4058  𝒫 cpw 4302   cint 4627  cmpt 4881  cfv 6049  Moorecmre 16464  mrClscmrc 16465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-mre 16468  df-mrc 16469
This theorem is referenced by:  mrcid  16495  mrcss  16498  mrcssid  16499  cycsubg2  17852  aspval2  19569
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