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Metamath Proof Explorer

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 )