Theorem dmun

Description: The domain of a union is the union of domains. Exercise 56(a) of Enderton p. 65. (Contributed by NM, 12-Aug-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	dmun	⊢ dom ( 𝐴 ∪ 𝐵 ) = ( dom 𝐴 ∪ dom 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		unab	⊢ ( { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 } ∪ { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 } ) = { 𝑦 ∣ ( ∃ 𝑥 𝑦 𝐴 𝑥 ∨ ∃ 𝑥 𝑦 𝐵 𝑥 ) }
2		brun	⊢ ( 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 ↔ ( 𝑦 𝐴 𝑥 ∨ 𝑦 𝐵 𝑥 ) )
3	2	exbii	⊢ ( ∃ 𝑥 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 ↔ ∃ 𝑥 ( 𝑦 𝐴 𝑥 ∨ 𝑦 𝐵 𝑥 ) )
4		19.43	⊢ ( ∃ 𝑥 ( 𝑦 𝐴 𝑥 ∨ 𝑦 𝐵 𝑥 ) ↔ ( ∃ 𝑥 𝑦 𝐴 𝑥 ∨ ∃ 𝑥 𝑦 𝐵 𝑥 ) )
5	3 4	bitr2i	⊢ ( ( ∃ 𝑥 𝑦 𝐴 𝑥 ∨ ∃ 𝑥 𝑦 𝐵 𝑥 ) ↔ ∃ 𝑥 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 )
6	5	abbii	⊢ { 𝑦 ∣ ( ∃ 𝑥 𝑦 𝐴 𝑥 ∨ ∃ 𝑥 𝑦 𝐵 𝑥 ) } = { 𝑦 ∣ ∃ 𝑥 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 }
7	1 6	eqtri	⊢ ( { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 } ∪ { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 } ) = { 𝑦 ∣ ∃ 𝑥 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 }
8		df-dm	⊢ dom 𝐴 = { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 }
9		df-dm	⊢ dom 𝐵 = { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 }
10	8 9	uneq12i	⊢ ( dom 𝐴 ∪ dom 𝐵 ) = ( { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 } ∪ { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 } )
11		df-dm	⊢ dom ( 𝐴 ∪ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 }
12	7 10 11	3eqtr4ri	⊢ dom ( 𝐴 ∪ 𝐵 ) = ( dom 𝐴 ∪ dom 𝐵 )