Theorem mrcun 17538

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 1136	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mre1cl 17506	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
3		elpw2g 5275	. . . . . . 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 477	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
6	5	3adant3 1132	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
7		elpw2g 5275	. . . . . . 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 477	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
10	9	3adant2 1131	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
11	6, 10	prssd 4775	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
12		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
13	12	mrcuni 17537	. . 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 4876	. . . 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 6835	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘(𝑈 ∪ 𝑉)))
18	12	mrcf 17525	. . . . . . . 8 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
19	18	ffnd 6660	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
20	19	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐹 Fn 𝒫 𝑋)
21		fnimapr 6914	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
22	20, 6, 10, 21	syl3anc 1373	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
23	22	unieqd 4873	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ∪ {(𝐹‘𝑈), (𝐹‘𝑉)})
24		fvex 6844	. . . . 5 ⊢ (𝐹‘𝑈) ∈ V
25		fvex 6844	. . . . 5 ⊢ (𝐹‘𝑉) ∈ V
26	24, 25	unipr 4877	. . . 4 ⊢ ∪ {(𝐹‘𝑈), (𝐹‘𝑉)} = ((𝐹‘𝑈) ∪ (𝐹‘𝑉))
27	23, 26	eqtrdi 2784	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ((𝐹‘𝑈) ∪ (𝐹‘𝑉)))
28	27	fveq2d 6835	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))
29	14, 17, 28	3eqtr3d 2776	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4551  {cpr 4579  ∪ cuni 4860   “ cima 5624   Fn wfn 6484  ‘cfv 6489  Moorecmre 17494  mrClscmrc 17495

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-mre 17498  df-mrc 17499

This theorem is referenced by: (None)