Theorem mrcun 17586

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 1142	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mre1cl 17554	. . . . . . 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 1138	. . . 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 1137	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
11	6, 10	prssd 4760	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
12		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
13	12	mrcuni 17585	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})))
14	1, 11, 13	syl2anc 590	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})))
15		uniprg 4861	. . . 4 ⊢ ((𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → ∪ {𝑈, 𝑉} = (𝑈 ∪ 𝑉))
16	6, 10, 15	syl2anc 590	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ {𝑈, 𝑉} = (𝑈 ∪ 𝑉))
17	16	fveq2d 6838	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘(𝑈 ∪ 𝑉)))
18	12	mrcf 17573	. . . . . . . 8 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
19	18	ffnd 6663	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
20	19	3ad2ant1 1139	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐹 Fn 𝒫 𝑋)
21		fnimapr 6917	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
22	20, 6, 10, 21	syl3anc 1379	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
23	22	unieqd 4858	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ∪ {(𝐹‘𝑈), (𝐹‘𝑉)})
24		fvex 6847	. . . . 5 ⊢ (𝐹‘𝑈) ∈ V
25		fvex 6847	. . . . 5 ⊢ (𝐹‘𝑉) ∈ V
26	24, 25	unipr 4862	. . . 4 ⊢ ∪ {(𝐹‘𝑈), (𝐹‘𝑉)} = ((𝐹‘𝑈) ∪ (𝐹‘𝑉))
27	23, 26	eqtrdi 2791	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ((𝐹‘𝑈) ∪ (𝐹‘𝑉)))
28	27	fveq2d 6838	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))
29	14, 17, 28	3eqtr3d 2783	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∪ cun 3888   ⊆ wss 3890  𝒫 cpw 4536  {cpr 4564  ∪ cuni 4845   “ cima 5628   Fn wfn 6487  ‘cfv 6492  Moorecmre 17542  mrClscmrc 17543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mre 17546  df-mrc 17547

This theorem is referenced by: (None)