Theorem mrcuni 17775

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 488	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		simpll 779	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
3		ssel2 3926	. . . . . . . . 9 ⊢ ((𝑈 ⊆ 𝒫 𝑋 ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝒫 𝑋)
4	3	elpwid 4566	. . . . . . . 8 ⊢ ((𝑈 ⊆ 𝒫 𝑋 ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ 𝑋)
5	4	adantll 727	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ 𝑋)
6		mrcfval.f	. . . . . . . 8 ⊢ 𝐹 = (mrCls‘𝐶)
7	6	mrcssid 17771	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ⊆ 𝑋) → 𝑠 ⊆ (𝐹‘𝑠))
8	2, 5, 7	syl2anc 596	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ (𝐹‘𝑠))
9	6	mrcf 17763	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
10	9	ffund 6706	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
11	10	adantr 486	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
12	9	fdmd 6712	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
13	12	sseq2d 3963	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹 ↔ 𝑈 ⊆ 𝒫 𝑋))
14	13	biimpar 483	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
15		funfvima2 7229	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑈 ⊆ dom 𝐹) → (𝑠 ∈ 𝑈 → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈)))
16	11, 14, 15	syl2anc 596	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠 ∈ 𝑈 → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈)))
17	16	imp 412	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → (𝐹‘𝑠) ∈ (𝐹 “ 𝑈))
18		elssuni 4899	. . . . . . 7 ⊢ ((𝐹‘𝑠) ∈ (𝐹 “ 𝑈) → (𝐹‘𝑠) ⊆ ∪ (𝐹 “ 𝑈))
19	17, 18	syl 18	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → (𝐹‘𝑠) ⊆ ∪ (𝐹 “ 𝑈))
20	8, 19	sstrd 3941	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
21	20	ralrimiva 3155	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ 𝑈 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
22		unissb 4901	. . . 4 ⊢ (∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈) ↔ ∀𝑠 ∈ 𝑈 𝑠 ⊆ ∪ (𝐹 “ 𝑈))
23	21, 22	sylibr 237	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈))
24	6	mrcssv 17768	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑥) ⊆ 𝑋)
25	24	adantr 486	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘𝑥) ⊆ 𝑋)
26	25	ralrimivw 3159	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋)
27	9	ffnd 6702	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
28		sseq1 3956	. . . . . . 7 ⊢ (𝑠 = (𝐹‘𝑥) → (𝑠 ⊆ 𝑋 ↔ (𝐹‘𝑥) ⊆ 𝑋))
29	28	ralima 7235	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋 ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋))
30	27, 29	sylan 592	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋 ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ 𝑋))
31	26, 30	mpbird 260	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋)
32		unissb 4901	. . . 4 ⊢ (∪ (𝐹 “ 𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ 𝑋)
33	31, 32	sylibr 237	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ (𝐹 “ 𝑈) ⊆ 𝑋)
34	6	mrcss 17770	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ ∪ (𝐹 “ 𝑈) ∧ ∪ (𝐹 “ 𝑈) ⊆ 𝑋) → (𝐹‘∪ 𝑈) ⊆ (𝐹‘∪ (𝐹 “ 𝑈)))
35	1, 23, 33, 34	syl3anc 1398	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) ⊆ (𝐹‘∪ (𝐹 “ 𝑈)))
36		simpll 779	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
37		elssuni 4899	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
38	37	adantl 487	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ ∪ 𝑈)
39		sspwuni 5060	. . . . . . . . . 10 ⊢ (𝑈 ⊆ 𝒫 𝑋 ↔ ∪ 𝑈 ⊆ 𝑋)
40	39	bilani 510	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ 𝑈 ⊆ 𝑋)
41	40	adantr 486	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → ∪ 𝑈 ⊆ 𝑋)
42	6	mrcss 17770	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ ∪ 𝑈 ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
43	36, 38, 41, 42	syl3anc 1398	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑈) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
44	43	ralrimiva 3155	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈))
45		sseq1 3956	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑥) → (𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
46	45	ralima 7235	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
47	27, 46	sylan 592	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑈)))
48	44, 47	mpbird 260	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈))
49		unissb 4901	. . . . 5 ⊢ (∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈) ↔ ∀𝑠 ∈ (𝐹 “ 𝑈)𝑠 ⊆ (𝐹‘∪ 𝑈))
50	48, 49	sylibr 237	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈))
51	6	mrcssv 17768	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘∪ 𝑈) ⊆ 𝑋)
52	51	adantr 486	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) ⊆ 𝑋)
53	6	mrcss 17770	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ (𝐹 “ 𝑈) ⊆ (𝐹‘∪ 𝑈) ∧ (𝐹‘∪ 𝑈) ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘(𝐹‘∪ 𝑈)))
54	1, 50, 52, 53	syl3anc 1398	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘(𝐹‘∪ 𝑈)))
55	6	mrcidm 17773	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘(𝐹‘∪ 𝑈)) = (𝐹‘∪ 𝑈))
56	1, 40, 55	syl2anc 596	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹‘∪ 𝑈)) = (𝐹‘∪ 𝑈))
57	54, 56	sseqtrd 3967	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ (𝐹 “ 𝑈)) ⊆ (𝐹‘∪ 𝑈))
58	35, 57	eqssd 3948	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) = (𝐹‘∪ (𝐹 “ 𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  dom cdm 5651   “ cima 5654  Fun wfun 6525   Fn wfn 6526  ‘cfv 6531  Moorecmre 17732  mrClscmrc 17733

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-mre 17736  df-mrc 17737

This theorem is used by:  mrcun  17776  isacs4lem  18698