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Theorem mrcflem 16872
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 486 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 ssrab2 4010 . . . 4 {𝑠𝐶𝑥𝑠} ⊆ 𝐶
32a1i 11 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ⊆ 𝐶)
4 sseq2 3944 . . . . 5 (𝑠 = 𝑋 → (𝑥𝑠𝑥𝑋))
5 mre1cl 16860 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
65adantr 484 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋𝐶)
7 elpwi 4509 . . . . . 6 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
87adantl 485 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
94, 6, 8elrabd 3633 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠𝐶𝑥𝑠})
109ne0d 4254 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ≠ ∅)
11 mreintcl 16861 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠𝐶𝑥𝑠} ⊆ 𝐶 ∧ {𝑠𝐶𝑥𝑠} ≠ ∅) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
121, 3, 10, 11syl3anc 1368 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
1312fmpttd 6860 1 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  wne 2990  {crab 3113  wss 3884  c0 4246  𝒫 cpw 4500   cint 4841  cmpt 5113  wf 6324  cfv 6328  Moorecmre 16848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-mre 16852
This theorem is referenced by:  fnmrc  16873  mrcfval  16874  mrcf  16875
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