Theorem fnmrc 17513

Description: Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fnmrc	⊢ mrCls Fn ∪ ran Moore

Proof of Theorem fnmrc

Dummy variables 𝑐 𝑥 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mrc 17489	. . 3 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
2	1	fnmpt 6622	. 2 ⊢ (∀𝑐 ∈ ∪ ran Moore(𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V → mrCls Fn ∪ ran Moore)
3		mreunirn 17503	. . 3 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
4		mrcflem 17512	. . . . 5 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
5		fssxp 6679	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐))
6	4, 5	syl 17	. . . 4 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐))
7		vuniex 7675	. . . . . 6 ⊢ ∪ 𝑐 ∈ V
8	7	pwex 5319	. . . . 5 ⊢ 𝒫 ∪ 𝑐 ∈ V
9		vex 3440	. . . . 5 ⊢ 𝑐 ∈ V
10	8, 9	xpex 7689	. . . 4 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
11		ssexg 5262	. . . 4 ⊢ (((𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐) ∧ (𝒫 ∪ 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
12	6, 10, 11	sylancl 586	. . 3 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
13	3, 12	sylbi 217	. 2 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
14	2, 13	mprg 3050	1 ⊢ mrCls Fn ∪ ran Moore

Syntax hints:   ∈ wcel 2109  {crab 3394  Vcvv 3436   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ∩ cint 4896   ↦ cmpt 5173   × cxp 5617  ran crn 5620   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  Moorecmre 17484  mrClscmrc 17485

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-mre 17488  df-mrc 17489

This theorem is referenced by:  ismrc  42694