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Metamath Proof Explorer

Theorem mptun

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

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

Proof

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