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Theorem mrcval 16871
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 16869 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
32adantr 481 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
4 sseq1 3996 . . . . 5 (𝑥 = 𝑈 → (𝑥𝑠𝑈𝑠))
54rabbidv 3486 . . . 4 (𝑥 = 𝑈 → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
65inteqd 4879 . . 3 (𝑥 = 𝑈 {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
76adantl 482 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) ∧ 𝑥 = 𝑈) → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
8 mre1cl 16855 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
9 elpw2g 5244 . . . 4 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
108, 9syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
1110biimpar 478 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
12 sseq2 3997 . . . . 5 (𝑠 = 𝑋 → (𝑈𝑠𝑈𝑋))
138adantr 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋𝐶)
14 simpr 485 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈𝑋)
1512, 13, 14elrabd 3686 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋 ∈ {𝑠𝐶𝑈𝑠})
1615ne0d 4305 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ≠ ∅)
17 intex 5237 . . 3 ({𝑠𝐶𝑈𝑠} ≠ ∅ ↔ {𝑠𝐶𝑈𝑠} ∈ V)
1816, 17sylib 219 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ∈ V)
193, 7, 11, 18fvmptd 6771 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  {crab 3147  Vcvv 3500  wss 3940  c0 4295  𝒫 cpw 4542   cint 4874  cmpt 5143  cfv 6352  Moorecmre 16843  mrClscmrc 16844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-mre 16847  df-mrc 16848
This theorem is referenced by:  mrcid  16874  mrcss  16877  mrcssid  16878  cycsubg2  18283  aspval2  20046
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