Metamath Proof Explorer


Theorem rfovfvfvd

Description: Value of the operator, ( A O B ) , which maps between relations and functions for relations between base sets, A and B , relation R , and left element X . (Contributed by RP, 25-Apr-2021)

Ref Expression
Hypotheses rfovd.rf 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑟 ∈ 𝒫 ( 𝑎 × 𝑏 ) ↦ ( 𝑥𝑎 ↦ { 𝑦𝑏𝑥 𝑟 𝑦 } ) ) )
rfovd.a ( 𝜑𝐴𝑉 )
rfovd.b ( 𝜑𝐵𝑊 )
rfovfvd.r ( 𝜑𝑅 ∈ 𝒫 ( 𝐴 × 𝐵 ) )
rfovfvd.f 𝐹 = ( 𝐴 𝑂 𝐵 )
rfovfvfvd.x ( 𝜑𝑋𝐴 )
rfovfvfvd.g 𝐺 = ( 𝐹𝑅 )
Assertion rfovfvfvd ( 𝜑 → ( 𝐺𝑋 ) = { 𝑦𝐵𝑋 𝑅 𝑦 } )

Proof

Step Hyp Ref Expression
1 rfovd.rf 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑟 ∈ 𝒫 ( 𝑎 × 𝑏 ) ↦ ( 𝑥𝑎 ↦ { 𝑦𝑏𝑥 𝑟 𝑦 } ) ) )
2 rfovd.a ( 𝜑𝐴𝑉 )
3 rfovd.b ( 𝜑𝐵𝑊 )
4 rfovfvd.r ( 𝜑𝑅 ∈ 𝒫 ( 𝐴 × 𝐵 ) )
5 rfovfvd.f 𝐹 = ( 𝐴 𝑂 𝐵 )
6 rfovfvfvd.x ( 𝜑𝑋𝐴 )
7 rfovfvfvd.g 𝐺 = ( 𝐹𝑅 )
8 1 2 3 4 5 rfovfvd ( 𝜑 → ( 𝐹𝑅 ) = ( 𝑥𝐴 ↦ { 𝑦𝐵𝑥 𝑅 𝑦 } ) )
9 7 8 syl5eq ( 𝜑𝐺 = ( 𝑥𝐴 ↦ { 𝑦𝐵𝑥 𝑅 𝑦 } ) )
10 breq1 ( 𝑥 = 𝑋 → ( 𝑥 𝑅 𝑦𝑋 𝑅 𝑦 ) )
11 10 rabbidv ( 𝑥 = 𝑋 → { 𝑦𝐵𝑥 𝑅 𝑦 } = { 𝑦𝐵𝑋 𝑅 𝑦 } )
12 11 adantl ( ( 𝜑𝑥 = 𝑋 ) → { 𝑦𝐵𝑥 𝑅 𝑦 } = { 𝑦𝐵𝑋 𝑅 𝑦 } )
13 rabexg ( 𝐵𝑊 → { 𝑦𝐵𝑋 𝑅 𝑦 } ∈ V )
14 3 13 syl ( 𝜑 → { 𝑦𝐵𝑋 𝑅 𝑦 } ∈ V )
15 9 12 6 14 fvmptd ( 𝜑 → ( 𝐺𝑋 ) = { 𝑦𝐵𝑋 𝑅 𝑦 } )