Metamath Proof Explorer


Theorem fsovfvfvd

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 and element Y . (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 ) )
fsovfvfvd.h 𝐻 = ( 𝐺𝐹 )
fsovfvfvd.y ( 𝜑𝑌𝐵 )
Assertion fsovfvfvd ( 𝜑 → ( 𝐻𝑌 ) = { 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 fsovfvfvd.h 𝐻 = ( 𝐺𝐹 )
7 fsovfvfvd.y ( 𝜑𝑌𝐵 )
8 1 2 3 4 5 fsovfvd ( 𝜑 → ( 𝐺𝐹 ) = ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) )
9 6 8 syl5eq ( 𝜑𝐻 = ( 𝑦𝐵 ↦ { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } ) )
10 eleq1 ( 𝑦 = 𝑌 → ( 𝑦 ∈ ( 𝐹𝑥 ) ↔ 𝑌 ∈ ( 𝐹𝑥 ) ) )
11 10 rabbidv ( 𝑦 = 𝑌 → { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } = { 𝑥𝐴𝑌 ∈ ( 𝐹𝑥 ) } )
12 11 adantl ( ( 𝜑𝑦 = 𝑌 ) → { 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) } = { 𝑥𝐴𝑌 ∈ ( 𝐹𝑥 ) } )
13 rabexg ( 𝐴𝑉 → { 𝑥𝐴𝑌 ∈ ( 𝐹𝑥 ) } ∈ V )
14 2 13 syl ( 𝜑 → { 𝑥𝐴𝑌 ∈ ( 𝐹𝑥 ) } ∈ V )
15 9 12 7 14 fvmptd ( 𝜑 → ( 𝐻𝑌 ) = { 𝑥𝐴𝑌 ∈ ( 𝐹𝑥 ) } )