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		breq1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝐴 𝑥 ↔ 𝑧 𝐴 𝑥 ) )
2	1	exbidv	⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑥 𝑦 𝐴 𝑥 ↔ ∃ 𝑥 𝑧 𝐴 𝑥 ) )
3		breq1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝐵 𝑥 ↔ 𝑧 𝐵 𝑥 ) )
4	3	exbidv	⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑥 𝑦 𝐵 𝑥 ↔ ∃ 𝑥 𝑧 𝐵 𝑥 ) )
5	2 4	unabw	⊢ ( { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 } ∪ { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 } ) = { 𝑧 ∣ ( ∃ 𝑥 𝑧 𝐴 𝑥 ∨ ∃ 𝑥 𝑧 𝐵 𝑥 ) }
6		brun	⊢ ( 𝑧 ( 𝐴 ∪ 𝐵 ) 𝑥 ↔ ( 𝑧 𝐴 𝑥 ∨ 𝑧 𝐵 𝑥 ) )
7	6	exbii	⊢ ( ∃ 𝑥 𝑧 ( 𝐴 ∪ 𝐵 ) 𝑥 ↔ ∃ 𝑥 ( 𝑧 𝐴 𝑥 ∨ 𝑧 𝐵 𝑥 ) )
8		19.43	⊢ ( ∃ 𝑥 ( 𝑧 𝐴 𝑥 ∨ 𝑧 𝐵 𝑥 ) ↔ ( ∃ 𝑥 𝑧 𝐴 𝑥 ∨ ∃ 𝑥 𝑧 𝐵 𝑥 ) )
9	7 8	bitr2i	⊢ ( ( ∃ 𝑥 𝑧 𝐴 𝑥 ∨ ∃ 𝑥 𝑧 𝐵 𝑥 ) ↔ ∃ 𝑥 𝑧 ( 𝐴 ∪ 𝐵 ) 𝑥 )
10	9	abbii	⊢ { 𝑧 ∣ ( ∃ 𝑥 𝑧 𝐴 𝑥 ∨ ∃ 𝑥 𝑧 𝐵 𝑥 ) } = { 𝑧 ∣ ∃ 𝑥 𝑧 ( 𝐴 ∪ 𝐵 ) 𝑥 }
11	5 10	eqtri	⊢ ( { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 } ∪ { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 } ) = { 𝑧 ∣ ∃ 𝑥 𝑧 ( 𝐴 ∪ 𝐵 ) 𝑥 }
12		df-dm	⊢ dom 𝐴 = { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 }
13		df-dm	⊢ dom 𝐵 = { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 }
14	12 13	uneq12i	⊢ ( dom 𝐴 ∪ dom 𝐵 ) = ( { 𝑦 ∣ ∃ 𝑥 𝑦 𝐴 𝑥 } ∪ { 𝑦 ∣ ∃ 𝑥 𝑦 𝐵 𝑥 } )
15		df-dm	⊢ dom ( 𝐴 ∪ 𝐵 ) = { 𝑧 ∣ ∃ 𝑥 𝑧 ( 𝐴 ∪ 𝐵 ) 𝑥 }
16	11 14 15	3eqtr4ri	⊢ dom ( 𝐴 ∪ 𝐵 ) = ( dom 𝐴 ∪ dom 𝐵 )