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Theorem mrcflem 16206
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 473 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 ssrab2 3672 . . . 4 {𝑠𝐶𝑥𝑠} ⊆ 𝐶
32a1i 11 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ⊆ 𝐶)
4 mre1cl 16194 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
54adantr 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋𝐶)
6 elpwi 4146 . . . . . 6 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
76adantl 482 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥𝑋)
8 sseq2 3612 . . . . . 6 (𝑠 = 𝑋 → (𝑥𝑠𝑥𝑋))
98elrab 3351 . . . . 5 (𝑋 ∈ {𝑠𝐶𝑥𝑠} ↔ (𝑋𝐶𝑥𝑋))
105, 7, 9sylanbrc 697 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠𝐶𝑥𝑠})
11 ne0i 3903 . . . 4 (𝑋 ∈ {𝑠𝐶𝑥𝑠} → {𝑠𝐶𝑥𝑠} ≠ ∅)
1210, 11syl 17 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ≠ ∅)
13 mreintcl 16195 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠𝐶𝑥𝑠} ⊆ 𝐶 ∧ {𝑠𝐶𝑥𝑠} ≠ ∅) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
141, 3, 12, 13syl3anc 1323 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠𝐶𝑥𝑠} ∈ 𝐶)
15 eqid 2621 . 2 (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠})
1614, 15fmptd 6351 1 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wne 2790  {crab 2912  wss 3560  c0 3897  𝒫 cpw 4136   cint 4447  cmpt 4683  wf 5853  cfv 5857  Moorecmre 16182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-mre 16186
This theorem is referenced by:  fnmrc  16207  mrcfval  16208  mrcf  16209
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