Theorem mrcuni 16281

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 473	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		simpll 790	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
3		ssel2 3598	. . . . . . . . 9 ⊢ ((𝑈 ⊆ 𝒫 𝑋 ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝒫 𝑋)
4	3	elpwid 4170	. . . . . . . 8 ⊢ ((𝑈 ⊆ 𝒫 𝑋 ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ 𝑋)
5	4	adantll 750	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ 𝑋)
6		mrcfval.f	. . . . . . . 8 ⊢ 𝐹 = (mrCls‘𝐶)
7	6	mrcssid 16277	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ⊆ 𝑋) → 𝑠 ⊆ (𝐹‘𝑠))
8	2, 5, 7	syl2anc 693	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ (𝐹‘𝑠))
9	6	mrcf 16269	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
10		ffun 6048	. . . . . . . . . . 11 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → Fun 𝐹)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
12	11	adantr 481	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
13		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → dom 𝐹 = 𝒫 𝑋)
14	9, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
15	14	sseq2d 3633	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹 ↔ 𝑈 ⊆ 𝒫 𝑋))
16	15	biimpar 502	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
17		funfvima2 6493	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑈 ⊆ dom 𝐹) → (𝑠 ∈ 𝑈 → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈)))
18	12, 16, 17	syl2anc 693	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠 ∈ 𝑈 → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈)))
19	18	imp 445	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈))
20		elssuni 4467	. . . . . . 7 ⊢ ((𝐹‘𝑠) ∈ (𝐹 “ 𝑈) → (𝐹‘𝑠) ⊆ ∪ (𝐹 “ 𝑈))
21	19, 20	syl 17	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → (𝐹‘𝑠) ⊆ ∪ (𝐹 “ 𝑈))
22	8, 21	sstrd 3613	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
23	22	ralrimiva 2966	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ 𝑈 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
24		unissb 4469	. . . 4 ⊢ (∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈) ↔ ∀𝑠 ∈ 𝑈 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
25	23, 24	sylibr 224	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈))
26	6	mrcssv 16274	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑥) ⊆ 𝑋)
27	26	adantr 481	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘𝑥) ⊆ 𝑋)
28	27	ralrimivw 2967	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋)
29		ffn 6045	. . . . . . 7 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → 𝐹 Fn 𝒫 𝑋)
30	9, 29	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
31		sseq1 3626	. . . . . . 7 ⊢ (𝑠 = (𝐹‘𝑥) → (𝑠 ⊆ 𝑋 ↔ (𝐹‘𝑥) ⊆ 𝑋))
32	31	ralima 6498	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋 ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋))
33	30, 32	sylan 488	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋 ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋))
34	28, 33	mpbird 247	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋)
35		unissb 4469	. . . 4 ⊢ (∪ (𝐹 “ 𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋)
36	34, 35	sylibr 224	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ (𝐹 “ 𝑈) ⊆ 𝑋)
37	6	mrcss 16276	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈) ∧ ∪ (𝐹 “ 𝑈) ⊆ 𝑋) → (𝐹‘∪ 𝑈) ⊆ (𝐹‘∪ (𝐹 “ 𝑈)))
38	1, 25, 36, 37	syl3anc 1326	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) ⊆ (𝐹‘∪ (𝐹 “ 𝑈)))
39		simpll 790	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
40		elssuni 4467	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
41	40	adantl 482	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ ∪ 𝑈)
42		sspwuni 4611	. . . . . . . . . . 11 ⊢ (𝑈 ⊆ 𝒫 𝑋 ↔ ∪ 𝑈 ⊆ 𝑋)
43	42	biimpi 206	. . . . . . . . . 10 ⊢ (𝑈 ⊆ 𝒫 𝑋 → ∪ 𝑈 ⊆ 𝑋)
44	43	adantl 482	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ 𝑈 ⊆ 𝑋)
45	44	adantr 481	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → ∪ 𝑈 ⊆ 𝑋)
46	6	mrcss 16276	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ ∪ 𝑈 ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
47	39, 41, 45, 46	syl3anc 1326	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
48	47	ralrimiva 2966	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
49		sseq1 3626	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑥) → (𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
50	49	ralima 6498	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
51	30, 50	sylan 488	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
52	48, 51	mpbird 247	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈))
53		unissb 4469	. . . . 5 ⊢ (∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈))
54	52, 53	sylibr 224	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈))
55	6	mrcssv 16274	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘∪ 𝑈) ⊆ 𝑋)
56	55	adantr 481	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) ⊆ 𝑋)
57	6	mrcss 16276	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈) ∧ (𝐹‘∪ 𝑈) ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘(𝐹‘∪ 𝑈)))
58	1, 54, 56, 57	syl3anc 1326	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘(𝐹‘∪ 𝑈)))
59	6	mrcidm 16279	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘(𝐹‘∪ 𝑈)) = (𝐹‘∪ 𝑈))
60	1, 44, 59	syl2anc 693	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹‘∪ 𝑈)) = (𝐹‘∪ 𝑈))
61	58, 60	sseqtrd 3641	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘∪ 𝑈))
62	38, 61	eqssd 3620	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) = (𝐹‘∪ (𝐹 “ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  dom cdm 5114   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  Moorecmre 16242  mrClscmrc 16243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by:  mrcun  16282  isacs4lem  17168