Theorem mrcflem 17649

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 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶)

Distinct variable groups:   𝑥,𝑠,𝐶   𝑥,𝑋,𝑠

Proof of Theorem mrcflem

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		ssrab2 4080	. . . 4 ⊢ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶
3	2	a1i 11	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶)
4		sseq2 4010	. . . . 5 ⊢ (𝑠 = 𝑋 → (𝑥 ⊆ 𝑠 ↔ 𝑥 ⊆ 𝑋))
5		mre1cl 17637	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
6	5	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐶)
7		elpwi 4607	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
8	7	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋)
9	4, 6, 8	elrabd 3694	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
10	9	ne0d 4342	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅)
11		mreintcl 17638	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶 ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶)
12	1, 3, 10, 11	syl3anc 1373	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶)
13	12	fmpttd 7135	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ≠ wne 2940  {crab 3436   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  ∩ cint 4946   ↦ cmpt 5225  ⟶wf 6557  ‘cfv 6561  Moorecmre 17625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629

This theorem is referenced by:  fnmrc  17650  mrcfval  17651  mrcf  17652