Theorem mrcun 17797

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 1154	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mre1cl 17765	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
3		elpw2g 5294	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
4	2, 3	syl 18	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
5	4	biimpar 483	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
6	5	3adant3 1150	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
7		elpw2g 5294	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝑉 ∈ 𝒫 𝑋 ↔ 𝑉 ⊆ 𝑋))
8	2, 7	syl 18	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑉 ∈ 𝒫 𝑋 ↔ 𝑉 ⊆ 𝑋))
9	8	biimpar 483	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
10	9	3adant2 1149	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝑉 ∈ 𝒫 𝑋)
11	6, 10	prssd 4782	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
12		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
13	12	mrcuni 17796	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})))
14	1, 11, 13	syl2anc 596	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})))
15		uniprg 4882	. . . 4 ⊢ ((𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → ∪ {𝑈, 𝑉} = (𝑈 ∪ 𝑉))
16	6, 10, 15	syl2anc 596	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ {𝑈, 𝑉} = (𝑈 ∪ 𝑉))
17	16	fveq2d 6889	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ {𝑈, 𝑉}) = (𝐹‘(𝑈 ∪ 𝑉)))
18	12	mrcf 17784	. . . . . . . 8 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
19	18	ffnd 6710	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
20	19	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → 𝐹 Fn 𝒫 𝑋)
21		fnimapr 6968	. . . . . 6 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ 𝑈 ∈ 𝒫 𝑋 ∧ 𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
22	20, 6, 10, 21	syl3anc 1398	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹‘𝑈), (𝐹‘𝑉)})
23	22	unieqd 4879	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ∪ {(𝐹‘𝑈), (𝐹‘𝑉)})
24		fvex 6898	. . . . 5 ⊢ (𝐹‘𝑈) ∈ V
25		fvex 6898	. . . . 5 ⊢ (𝐹‘𝑉) ∈ V
26	24, 25	unipr 4883	. . . 4 ⊢ ∪ {(𝐹‘𝑈), (𝐹‘𝑉)} = ((𝐹‘𝑈) ∪ (𝐹‘𝑉))
27	23, 26	eqtrdi 2811	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → ∪ (𝐹 “ {𝑈, 𝑉}) = ((𝐹‘𝑈) ∪ (𝐹‘𝑉)))
28	27	fveq2d 6889	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘∪ (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))
29	14, 17, 28	3eqtr3d 2803	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  𝒫 cpw 4556  {cpr 4585  ∪ cuni 4866   “ cima 5653   Fn wfn 6533  ‘cfv 6538  Moorecmre 17753  mrClscmrc 17754

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-mre 17757  df-mrc 17758

This theorem is used by: (None)