Metamath Proof Explorer


Theorem dssmapfv2d

Description: Value of the duality operator for self-mappings of subsets of a base set, B when applied to function F . (Contributed by RP, 19-Apr-2021)

Ref Expression
Hypotheses dssmapfvd.o 𝑂 = ( 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏m 𝒫 𝑏 ) ↦ ( 𝑠 ∈ 𝒫 𝑏 ↦ ( 𝑏 ∖ ( 𝑓 ‘ ( 𝑏𝑠 ) ) ) ) ) )
dssmapfvd.d 𝐷 = ( 𝑂𝐵 )
dssmapfvd.b ( 𝜑𝐵𝑉 )
dssmapfv2d.f ( 𝜑𝐹 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) )
dssmapfv2d.g 𝐺 = ( 𝐷𝐹 )
Assertion dssmapfv2d ( 𝜑𝐺 = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 dssmapfvd.o 𝑂 = ( 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏m 𝒫 𝑏 ) ↦ ( 𝑠 ∈ 𝒫 𝑏 ↦ ( 𝑏 ∖ ( 𝑓 ‘ ( 𝑏𝑠 ) ) ) ) ) )
2 dssmapfvd.d 𝐷 = ( 𝑂𝐵 )
3 dssmapfvd.b ( 𝜑𝐵𝑉 )
4 dssmapfv2d.f ( 𝜑𝐹 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) )
5 dssmapfv2d.g 𝐺 = ( 𝐷𝐹 )
6 1 2 3 dssmapfvd ( 𝜑𝐷 = ( 𝑓 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) ↦ ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵𝑠 ) ) ) ) ) )
7 fveq1 ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝐵𝑠 ) ) = ( 𝐹 ‘ ( 𝐵𝑠 ) ) )
8 7 difeq2d ( 𝑓 = 𝐹 → ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵𝑠 ) ) ) = ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) )
9 8 mpteq2dv ( 𝑓 = 𝐹 → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵𝑠 ) ) ) ) = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) )
10 9 adantl ( ( 𝜑𝑓 = 𝐹 ) → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵𝑠 ) ) ) ) = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) )
11 pwexg ( 𝐵𝑉 → 𝒫 𝐵 ∈ V )
12 mptexg ( 𝒫 𝐵 ∈ V → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) ∈ V )
13 3 11 12 3syl ( 𝜑 → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) ∈ V )
14 6 10 4 13 fvmptd ( 𝜑 → ( 𝐷𝐹 ) = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) )
15 5 14 syl5eq ( 𝜑𝐺 = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵𝑠 ) ) ) ) )