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Theorem mrcflem 17650
Description: The domain and codomain 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 4089 . . . 4 {𝑠𝐶𝑥𝑠} ⊆ 𝐶
32a1i 11 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ⊆ 𝐶)
4 sseq2 4021 . . . . 5 (𝑠 = 𝑋 → (𝑥𝑠𝑥𝑋))
5 mre1cl 17638 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
65adantr 480 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋𝐶)
7 elpwi 4611 . . . . . 6 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
87adantl 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
94, 6, 8elrabd 3696 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠𝐶𝑥𝑠})
109ne0d 4347 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ≠ ∅)
11 mreintcl 17639 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠𝐶𝑥𝑠} ⊆ 𝐶 ∧ {𝑠𝐶𝑥𝑠} ≠ ∅) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
121, 3, 10, 11syl3anc 1370 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
1312fmpttd 7134 1 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wne 2937  {crab 3432  wss 3962  c0 4338  𝒫 cpw 4604   cint 4950  cmpt 5230  wf 6558  cfv 6562  Moorecmre 17626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mre 17630
This theorem is referenced by:  fnmrc  17651  mrcfval  17652  mrcf  17653
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