Theorem fnmrc 17807

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 17783	. . 3 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
2	1	fnmpt 6698	. 2 ⊢ (∀𝑐 ∈ ∪ ran Moore(𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V → mrCls Fn ∪ ran Moore)
3		mreunirn 17797	. . 3 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
4		mrcflem 17806	. . . . 5 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
5		fssxp 6756	. . . . 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 18	. . . 4 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐))
7		vuniex 7777	. . . . . 6 ⊢ ∪ 𝑐 ∈ V
8	7	pwex 5346	. . . . 5 ⊢ 𝒫 ∪ 𝑐 ∈ V
9		vex 3458	. . . . 5 ⊢ 𝑐 ∈ V
10	8, 9	xpex 7788	. . . 4 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
11		ssexg 5285	. . . 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 598	. . 3 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
13	3, 12	sylbi 220	. 2 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
14	2, 13	mprg 3086	1 ⊢ mrCls Fn ∪ ran Moore

This proof depends on syntax axioms:   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910   ↦ cmpt 5189   × cxp 5653  ran crn 5656   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  Moorecmre 17778  mrClscmrc 17779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-mre 17782  df-mrc 17783

This theorem is used by:  ismrc  43903