Metamath Proof Explorer


Theorem brovmptimex

Description: If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021)

Ref Expression
Hypotheses brovmptimex.mpt 𝐹 = ( 𝑥𝐸 , 𝑦𝐺𝐻 )
brovmptimex.br ( 𝜑𝐴 𝑅 𝐵 )
brovmptimex.ov ( 𝜑𝑅 = ( 𝐶 𝐹 𝐷 ) )
Assertion brovmptimex ( 𝜑 → ( 𝐶 ∈ V ∧ 𝐷 ∈ V ) )

Proof

Step Hyp Ref Expression
1 brovmptimex.mpt 𝐹 = ( 𝑥𝐸 , 𝑦𝐺𝐻 )
2 brovmptimex.br ( 𝜑𝐴 𝑅 𝐵 )
3 brovmptimex.ov ( 𝜑𝑅 = ( 𝐶 𝐹 𝐷 ) )
4 3 2 breqdi ( 𝜑𝐴 ( 𝐶 𝐹 𝐷 ) 𝐵 )
5 brne0 ( 𝐴 ( 𝐶 𝐹 𝐷 ) 𝐵 → ( 𝐶 𝐹 𝐷 ) ≠ ∅ )
6 1 reldmmpo Rel dom 𝐹
7 6 ovprc ( ¬ ( 𝐶 ∈ V ∧ 𝐷 ∈ V ) → ( 𝐶 𝐹 𝐷 ) = ∅ )
8 7 necon1ai ( ( 𝐶 𝐹 𝐷 ) ≠ ∅ → ( 𝐶 ∈ V ∧ 𝐷 ∈ V ) )
9 4 5 8 3syl ( 𝜑 → ( 𝐶 ∈ V ∧ 𝐷 ∈ V ) )