Theorem mrcun 17331

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

Hypothesis

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

Assertion

Ref	Expression
mrcun	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))

Proof of Theorem mrcun

Step	Hyp	Ref	Expression
1		simp1 1135	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mre1cl 17303	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
3		elpw2g 5268	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
5	4	biimpar 478	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
6	5	3adant3 1131	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
7		elpw2g 5268	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝑉 ∈ 𝒫 𝑋 ↔ 𝑉 ⊆ 𝑋))
8	2, 7	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑉 ∈ 𝒫 𝑋 ↔ 𝑉 ⊆ 𝑋))
9	8	biimpar 478	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
10	9	3adant2 1130	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
11	6, 10	prssd 4755	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
12		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
13	12	mrcuni 17330	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})))
14	1, 11, 13	syl2anc 584	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})))
15		uniprg 4856	. . . 4 ⊢ ((𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → ∪ {𝑈, 𝑉} = (𝑈 ∪ 𝑉))
16	6, 10, 15	syl2anc 584	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ {𝑈, 𝑉} = (𝑈 ∪ 𝑉))
17	16	fveq2d 6778	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘(𝑈 ∪ 𝑉)))
18	12	mrcf 17318	. . . . . . . 8 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
19	18	ffnd 6601	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
20	19	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐹 Fn 𝒫 𝑋)
21		fnimapr 6852	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
22	20, 6, 10, 21	syl3anc 1370	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
23	22	unieqd 4853	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ∪ {(𝐹‘𝑈), (𝐹‘𝑉)})
24		fvex 6787	. . . . 5 ⊢ (𝐹‘𝑈) ∈ V
25		fvex 6787	. . . . 5 ⊢ (𝐹‘𝑉) ∈ V
26	24, 25	unipr 4857	. . . 4 ⊢ ∪ {(𝐹‘𝑈), (𝐹‘𝑉)} = ((𝐹‘𝑈) ∪ (𝐹‘𝑉))
27	23, 26	eqtrdi 2794	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ((𝐹‘𝑈) ∪ (𝐹‘𝑉)))
28	27	fveq2d 6778	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))
29	14, 17, 28	3eqtr3d 2786	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  𝒫 cpw 4533  {cpr 4563  ∪ cuni 4839   “ cima 5592   Fn wfn 6428  ‘cfv 6433  Moorecmre 17291  mrClscmrc 17292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-mre 17295  df-mrc 17296

This theorem is referenced by: (None)