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Theorem mrcflem 16474
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 468 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 ssrab2 3836 . . . 4 {𝑠𝐶𝑥𝑠} ⊆ 𝐶
32a1i 11 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ⊆ 𝐶)
4 mre1cl 16462 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
54adantr 466 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋𝐶)
6 elpwi 4308 . . . . . 6 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
76adantl 467 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
8 sseq2 3776 . . . . . 6 (𝑠 = 𝑋 → (𝑥𝑠𝑥𝑋))
98elrab 3515 . . . . 5 (𝑋 ∈ {𝑠𝐶𝑥𝑠} ↔ (𝑋𝐶𝑥𝑋))
105, 7, 9sylanbrc 572 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠𝐶𝑥𝑠})
1110ne0d 4070 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ≠ ∅)
12 mreintcl 16463 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠𝐶𝑥𝑠} ⊆ 𝐶 ∧ {𝑠𝐶𝑥𝑠} ≠ ∅) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
131, 3, 11, 12syl3anc 1476 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
1413fmpttd 6530 1 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wne 2943  {crab 3065  wss 3723  c0 4063  𝒫 cpw 4298   cint 4612  cmpt 4864  wf 6026  cfv 6030  Moorecmre 16450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-mre 16454
This theorem is referenced by:  fnmrc  16475  mrcfval  16476  mrcf  16477
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