Metamath Proof Explorer


Theorem fsovfvd

Description: Value of the operator, ( A O B ) , which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B , when applied to function F . (Contributed by RP, 25-Apr-2021)

Ref Expression
Hypotheses fsovd.fs 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏m 𝑎 ) ↦ ( 𝑦𝑏 ↦ { 𝑥𝑎𝑦 ∈ ( 𝑓𝑥 ) } ) ) )
fsovd.a ( 𝜑𝐴𝑉 )
fsovd.b ( 𝜑𝐵𝑊 )
fsovfvd.g 𝐺 = ( 𝐴 𝑂 𝐵 )
fsovfvd.f ( 𝜑𝐹 ∈ ( 𝒫 𝐵m 𝐴 ) )
Assertion fsovfvd ( 𝜑 → ( 𝐺𝐹 ) = ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) )

Proof

Step Hyp Ref Expression
1 fsovd.fs 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏m 𝑎 ) ↦ ( 𝑦𝑏 ↦ { 𝑥𝑎𝑦 ∈ ( 𝑓𝑥 ) } ) ) )
2 fsovd.a ( 𝜑𝐴𝑉 )
3 fsovd.b ( 𝜑𝐵𝑊 )
4 fsovfvd.g 𝐺 = ( 𝐴 𝑂 𝐵 )
5 fsovfvd.f ( 𝜑𝐹 ∈ ( 𝒫 𝐵m 𝐴 ) )
6 1 2 3 fsovd ( 𝜑 → ( 𝐴 𝑂 𝐵 ) = ( 𝑓 ∈ ( 𝒫 𝐵m 𝐴 ) ↦ ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝑓𝑥 ) } ) ) )
7 4 6 syl5eq ( 𝜑𝐺 = ( 𝑓 ∈ ( 𝒫 𝐵m 𝐴 ) ↦ ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝑓𝑥 ) } ) ) )
8 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑥 ) = ( 𝐹𝑥 ) )
9 8 eleq2d ( 𝑓 = 𝐹 → ( 𝑦 ∈ ( 𝑓𝑥 ) ↔ 𝑦 ∈ ( 𝐹𝑥 ) ) )
10 9 rabbidv ( 𝑓 = 𝐹 → { 𝑥𝐴𝑦 ∈ ( 𝑓𝑥 ) } = { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } )
11 10 mpteq2dv ( 𝑓 = 𝐹 → ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝑓𝑥 ) } ) = ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) )
12 11 adantl ( ( 𝜑𝑓 = 𝐹 ) → ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝑓𝑥 ) } ) = ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) )
13 3 mptexd ( 𝜑 → ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) ∈ V )
14 7 12 5 13 fvmptd ( 𝜑 → ( 𝐺𝐹 ) = ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) )