Metamath Proof Explorer


Theorem unrab

Description: Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004)

Ref Expression
Assertion unrab ( { 𝑥𝐴𝜑 } ∪ { 𝑥𝐴𝜓 } ) = { 𝑥𝐴 ∣ ( 𝜑𝜓 ) }

Proof

Step Hyp Ref Expression
1 df-rab { 𝑥𝐴𝜑 } = { 𝑥 ∣ ( 𝑥𝐴𝜑 ) }
2 df-rab { 𝑥𝐴𝜓 } = { 𝑥 ∣ ( 𝑥𝐴𝜓 ) }
3 1 2 uneq12i ( { 𝑥𝐴𝜑 } ∪ { 𝑥𝐴𝜓 } ) = ( { 𝑥 ∣ ( 𝑥𝐴𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥𝐴𝜓 ) } )
4 df-rab { 𝑥𝐴 ∣ ( 𝜑𝜓 ) } = { 𝑥 ∣ ( 𝑥𝐴 ∧ ( 𝜑𝜓 ) ) }
5 unab ( { 𝑥 ∣ ( 𝑥𝐴𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥𝐴𝜓 ) } ) = { 𝑥 ∣ ( ( 𝑥𝐴𝜑 ) ∨ ( 𝑥𝐴𝜓 ) ) }
6 andi ( ( 𝑥𝐴 ∧ ( 𝜑𝜓 ) ) ↔ ( ( 𝑥𝐴𝜑 ) ∨ ( 𝑥𝐴𝜓 ) ) )
7 6 abbii { 𝑥 ∣ ( 𝑥𝐴 ∧ ( 𝜑𝜓 ) ) } = { 𝑥 ∣ ( ( 𝑥𝐴𝜑 ) ∨ ( 𝑥𝐴𝜓 ) ) }
8 5 7 eqtr4i ( { 𝑥 ∣ ( 𝑥𝐴𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥𝐴𝜓 ) } ) = { 𝑥 ∣ ( 𝑥𝐴 ∧ ( 𝜑𝜓 ) ) }
9 4 8 eqtr4i { 𝑥𝐴 ∣ ( 𝜑𝜓 ) } = ( { 𝑥 ∣ ( 𝑥𝐴𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥𝐴𝜓 ) } )
10 3 9 eqtr4i ( { 𝑥𝐴𝜑 } ∪ { 𝑥𝐴𝜓 } ) = { 𝑥𝐴 ∣ ( 𝜑𝜓 ) }