Theorem mptun 6696

Description: Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
mptun	⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ∪ (𝑥 ∈ 𝐵 ↦ 𝐶))

Proof of Theorem mptun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 5232	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 = 𝐶)}
2		df-mpt 5232	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
3		df-mpt 5232	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}
4	2, 3	uneq12i 4161	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∪ (𝑥 ∈ 𝐵 ↦ 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)})
5		elun 4148	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
6	5	anbi1i 623	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 = 𝐶))
7		andir 1006	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)))
8	6, 7	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)))
9	8	opabbii 5215	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 = 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶))}
10		unopab 5230	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶))}
11	9, 10	eqtr4i 2762	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 = 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)})
12	4, 11	eqtr4i 2762	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∪ (𝑥 ∈ 𝐵 ↦ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 = 𝐶)}
13	1, 12	eqtr4i 2762	1 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ∪ (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105   ∪ cun 3946  {copab 5210   ↦ cmpt 5231

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-opab 5211  df-mpt 5232

This theorem is referenced by:  partfun  6697  fmptap  7170  fmptapd  7171  esumrnmpt2  33530  ptrest  36951  fsuppssind  41628