Theorem cnvun

Description: The converse of a union is the union of converses. Theorem 16 of Suppes p. 62. (Contributed by NM, 25-Mar-1998) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	cnvun	⊢ ◡ ( 𝐴 ∪ 𝐵 ) = ( ◡ 𝐴 ∪ ◡ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-cnv	⊢ ◡ ( 𝐴 ∪ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 }
2		unopab	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐴 𝑥 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐵 𝑥 } ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 𝐴 𝑥 ∨ 𝑦 𝐵 𝑥 ) }
3		brun	⊢ ( 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 ↔ ( 𝑦 𝐴 𝑥 ∨ 𝑦 𝐵 𝑥 ) )
4	3	opabbii	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 𝐴 𝑥 ∨ 𝑦 𝐵 𝑥 ) }
5	2 4	eqtr4i	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐴 𝑥 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐵 𝑥 } ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 ( 𝐴 ∪ 𝐵 ) 𝑥 }
6	1 5	eqtr4i	⊢ ◡ ( 𝐴 ∪ 𝐵 ) = ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐴 𝑥 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐵 𝑥 } )
7		df-cnv	⊢ ◡ 𝐴 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐴 𝑥 }
8		df-cnv	⊢ ◡ 𝐵 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐵 𝑥 }
9	7 8	uneq12i	⊢ ( ◡ 𝐴 ∪ ◡ 𝐵 ) = ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐴 𝑥 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 𝐵 𝑥 } )
10	6 9	eqtr4i	⊢ ◡ ( 𝐴 ∪ 𝐵 ) = ( ◡ 𝐴 ∪ ◡ 𝐵 )