Theorem fnmrc 17652

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 17632	. . 3 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
2	1	fnmpt 6709	. 2 ⊢ (∀𝑐 ∈ ∪ ran Moore(𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V → mrCls Fn ∪ ran Moore)
3		mreunirn 17646	. . 3 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
4		mrcflem 17651	. . . . 5 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
5		fssxp 6764	. . . . 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 7758	. . . . . 6 ⊢ ∪ 𝑐 ∈ V
8	7	pwex 5386	. . . . 5 ⊢ 𝒫 ∪ 𝑐 ∈ V
9		vex 3482	. . . . 5 ⊢ 𝑐 ∈ V
10	8, 9	xpex 7772	. . . 4 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
11		ssexg 5329	. . . 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 3065	1 ⊢ mrCls Fn ∪ ran Moore

Syntax hints:   ∈ wcel 2106  {crab 3433  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  ∩ cint 4951   ↦ cmpt 5231   × cxp 5687  ran crn 5690   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  Moorecmre 17627  mrClscmrc 17628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mre 17631  df-mrc 17632

This theorem is referenced by:  ismrc  42689