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 ) )