Metamath Proof Explorer


Theorem sprsymrelfv

Description: The value of the function F which maps a subset of the set of pairs over a fixed set V to the relation relating two elements of the set V iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021)

Ref Expression
Hypotheses sprsymrelf.p 𝑃 = 𝒫 ( Pairs ‘ 𝑉 )
sprsymrelf.r 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥𝑉𝑦𝑉 ( 𝑥 𝑟 𝑦𝑦 𝑟 𝑥 ) }
sprsymrelf.f 𝐹 = ( 𝑝𝑃 ↦ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑝 𝑐 = { 𝑥 , 𝑦 } } )
Assertion sprsymrelfv ( 𝑋𝑃 → ( 𝐹𝑋 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑋 𝑐 = { 𝑥 , 𝑦 } } )

Proof

Step Hyp Ref Expression
1 sprsymrelf.p 𝑃 = 𝒫 ( Pairs ‘ 𝑉 )
2 sprsymrelf.r 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥𝑉𝑦𝑉 ( 𝑥 𝑟 𝑦𝑦 𝑟 𝑥 ) }
3 sprsymrelf.f 𝐹 = ( 𝑝𝑃 ↦ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑝 𝑐 = { 𝑥 , 𝑦 } } )
4 rexeq ( 𝑝 = 𝑋 → ( ∃ 𝑐𝑝 𝑐 = { 𝑥 , 𝑦 } ↔ ∃ 𝑐𝑋 𝑐 = { 𝑥 , 𝑦 } ) )
5 4 opabbidv ( 𝑝 = 𝑋 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑝 𝑐 = { 𝑥 , 𝑦 } } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑋 𝑐 = { 𝑥 , 𝑦 } } )
6 id ( 𝑋𝑃𝑋𝑃 )
7 elpwi ( 𝑋 ∈ 𝒫 ( Pairs ‘ 𝑉 ) → 𝑋 ⊆ ( Pairs ‘ 𝑉 ) )
8 7 1 eleq2s ( 𝑋𝑃𝑋 ⊆ ( Pairs ‘ 𝑉 ) )
9 sprsymrelfvlem ( 𝑋 ⊆ ( Pairs ‘ 𝑉 ) → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑋 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) )
10 8 9 syl ( 𝑋𝑃 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑋 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) )
11 3 5 6 10 fvmptd3 ( 𝑋𝑃 → ( 𝐹𝑋 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑐𝑋 𝑐 = { 𝑥 , 𝑦 } } )