This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem pwmndgplus

Description: The operation of the monoid of the power set of a class A under union. (Contributed by AV, 27-Feb-2024)

Ref Expression
Hypotheses pwmnd.b ⊢ ( Base ‘ 𝑀 ) = 𝒫 𝐴
pwmnd.p ⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝒫 𝐴 , 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝑥 ∪ 𝑦 ) )
Assertion pwmndgplus ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) → ( 𝑋 ( +g ‘ 𝑀 ) 𝑌 ) = ( 𝑋 ∪ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 pwmnd.b ⊢ ( Base ‘ 𝑀 ) = 𝒫 𝐴
2 pwmnd.p ⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝒫 𝐴 , 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝑥 ∪ 𝑦 ) )
3 2 a1i ⊢ ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) → ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝒫 𝐴 , 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝑥 ∪ 𝑦 ) ) )
4 uneq12 ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 ∪ 𝑦 ) = ( 𝑋 ∪ 𝑌 ) )
5 4 adantl ⊢ ( ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 ∪ 𝑦 ) = ( 𝑋 ∪ 𝑌 ) )
6 simpl ⊢ ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) → 𝑋 ∈ 𝒫 𝐴 )
7 simpr ⊢ ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) → 𝑌 ∈ 𝒫 𝐴 )
8 unexg ⊢ ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) → ( 𝑋 ∪ 𝑌 ) ∈ V )
9 3 5 6 7 8 ovmpod ⊢ ( ( 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ) → ( 𝑋 ( +g ‘ 𝑀 ) 𝑌 ) = ( 𝑋 ∪ 𝑌 ) )