Theorem mrcflem 17572

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 4020	. . . 4 ⊢ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶
3	2	a1i 11	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶)
4		sseq2 3948	. . . . 5 ⊢ (𝑠 = 𝑋 → (𝑥 ⊆ 𝑠 ↔ 𝑥 ⊆ 𝑋))
5		mre1cl 17556	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
6	5	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐶)
7		elpwi 4548	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
8	7	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋)
9	4, 6, 8	elrabd 3636	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
10	9	ne0d 4282	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅)
11		mreintcl 17557	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶 ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶)
12	1, 3, 10, 11	syl3anc 1374	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶)
13	12	fmpttd 7067	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ≠ wne 2932  {crab 3389   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∩ cint 4889   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  Moorecmre 17544

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-mre 17548

This theorem is referenced by:  fnmrc  17573  mrcfval  17574  mrcf  17575