Theorem mrcuni 17524

Description: Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
mrcuni	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) = (𝐹‘∪ (𝐹 “ 𝑈)))

Proof of Theorem mrcuni

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

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		simpll 766	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
3		ssel2 3929	. . . . . . . . 9 ⊢ ((𝑈 ⊆ 𝒫 𝑋 ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝒫 𝑋)
4	3	elpwid 4559	. . . . . . . 8 ⊢ ((𝑈 ⊆ 𝒫 𝑋 ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ 𝑋)
5	4	adantll 714	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ 𝑋)
6		mrcfval.f	. . . . . . . 8 ⊢ 𝐹 = (mrCls‘𝐶)
7	6	mrcssid 17520	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ⊆ 𝑋) → 𝑠 ⊆ (𝐹‘𝑠))
8	2, 5, 7	syl2anc 584	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ (𝐹‘𝑠))
9	6	mrcf 17512	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
10	9	ffund 6655	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
12	9	fdmd 6661	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
13	12	sseq2d 3967	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹 ↔ 𝑈 ⊆ 𝒫 𝑋))
14	13	biimpar 477	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
15		funfvima2 7165	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑈 ⊆ dom 𝐹) → (𝑠 ∈ 𝑈 → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈)))
16	11, 14, 15	syl2anc 584	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠 ∈ 𝑈 → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈)))
17	16	imp 406	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈))
18		elssuni 4889	. . . . . . 7 ⊢ ((𝐹‘𝑠) ∈ (𝐹 “ 𝑈) → (𝐹‘𝑠) ⊆ ∪ (𝐹 “ 𝑈))
19	17, 18	syl 17	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → (𝐹‘𝑠) ⊆ ∪ (𝐹 “ 𝑈))
20	8, 19	sstrd 3945	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
21	20	ralrimiva 3124	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ 𝑈 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
22		unissb 4891	. . . 4 ⊢ (∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈) ↔ ∀𝑠 ∈ 𝑈 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
23	21, 22	sylibr 234	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈))
24	6	mrcssv 17517	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑥) ⊆ 𝑋)
25	24	adantr 480	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘𝑥) ⊆ 𝑋)
26	25	ralrimivw 3128	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋)
27	9	ffnd 6652	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
28		sseq1 3960	. . . . . . 7 ⊢ (𝑠 = (𝐹‘𝑥) → (𝑠 ⊆ 𝑋 ↔ (𝐹‘𝑥) ⊆ 𝑋))
29	28	ralima 7171	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋 ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋))
30	27, 29	sylan 580	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋 ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋))
31	26, 30	mpbird 257	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋)
32		unissb 4891	. . . 4 ⊢ (∪ (𝐹 “ 𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋)
33	31, 32	sylibr 234	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ (𝐹 “ 𝑈) ⊆ 𝑋)
34	6	mrcss 17519	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈) ∧ ∪ (𝐹 “ 𝑈) ⊆ 𝑋) → (𝐹‘∪ 𝑈) ⊆ (𝐹‘∪ (𝐹 “ 𝑈)))
35	1, 23, 33, 34	syl3anc 1373	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) ⊆ (𝐹‘∪ (𝐹 “ 𝑈)))
36		simpll 766	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
37		elssuni 4889	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
38	37	adantl 481	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ ∪ 𝑈)
39		sspwuni 5048	. . . . . . . . . . 11 ⊢ (𝑈 ⊆ 𝒫 𝑋 ↔ ∪ 𝑈 ⊆ 𝑋)
40	39	biimpi 216	. . . . . . . . . 10 ⊢ (𝑈 ⊆ 𝒫 𝑋 → ∪ 𝑈 ⊆ 𝑋)
41	40	adantl 481	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ 𝑈 ⊆ 𝑋)
42	41	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → ∪ 𝑈 ⊆ 𝑋)
43	6	mrcss 17519	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ ∪ 𝑈 ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
44	36, 38, 42, 43	syl3anc 1373	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
45	44	ralrimiva 3124	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
46		sseq1 3960	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑥) → (𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
47	46	ralima 7171	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
48	27, 47	sylan 580	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
49	45, 48	mpbird 257	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈))
50		unissb 4891	. . . . 5 ⊢ (∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈))
51	49, 50	sylibr 234	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈))
52	6	mrcssv 17517	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘∪ 𝑈) ⊆ 𝑋)
53	52	adantr 480	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) ⊆ 𝑋)
54	6	mrcss 17519	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈) ∧ (𝐹‘∪ 𝑈) ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘(𝐹‘∪ 𝑈)))
55	1, 51, 53, 54	syl3anc 1373	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘(𝐹‘∪ 𝑈)))
56	6	mrcidm 17522	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘(𝐹‘∪ 𝑈)) = (𝐹‘∪ 𝑈))
57	1, 41, 56	syl2anc 584	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹‘∪ 𝑈)) = (𝐹‘∪ 𝑈))
58	55, 57	sseqtrd 3971	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘∪ 𝑈))
59	35, 58	eqssd 3952	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) = (𝐹‘∪ (𝐹 “ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3902  𝒫 cpw 4550  ∪ cuni 4859  dom cdm 5616   “ cima 5619  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481  Moorecmre 17481  mrClscmrc 17482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-mre 17485  df-mrc 17486

This theorem is referenced by:  mrcun  17525  isacs4lem  18447