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Theorem mrcflem 17232
Description: The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Assertion
Ref Expression
mrcflem (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
Distinct variable groups:   𝑥,𝑠,𝐶   𝑥,𝑋,𝑠

Proof of Theorem mrcflem
StepHypRef Expression
1 simpl 482 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 ssrab2 4009 . . . 4 {𝑠𝐶𝑥𝑠} ⊆ 𝐶
32a1i 11 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ⊆ 𝐶)
4 sseq2 3943 . . . . 5 (𝑠 = 𝑋 → (𝑥𝑠𝑥𝑋))
5 mre1cl 17220 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
65adantr 480 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋𝐶)
7 elpwi 4539 . . . . . 6 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
87adantl 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
94, 6, 8elrabd 3619 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠𝐶𝑥𝑠})
109ne0d 4266 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ≠ ∅)
11 mreintcl 17221 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠𝐶𝑥𝑠} ⊆ 𝐶 ∧ {𝑠𝐶𝑥𝑠} ≠ ∅) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
121, 3, 10, 11syl3anc 1369 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
1312fmpttd 6971 1 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wne 2942  {crab 3067  wss 3883  c0 4253  𝒫 cpw 4530   cint 4876  cmpt 5153  wf 6414  cfv 6418  Moorecmre 17208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-mre 17212
This theorem is referenced by:  fnmrc  17233  mrcfval  17234  mrcf  17235
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