Theorem elmade2

Description: Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024)

	Ref	Expression
Assertion	elmade2	⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ( O ‘ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( O ‘ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elmade	⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
2		oldval	⊢ ( 𝐴 ∈ On → ( O ‘ 𝐴 ) = ∪ ( M “ 𝐴 ) )
3	2	pweqd	⊢ ( 𝐴 ∈ On → 𝒫 ( O ‘ 𝐴 ) = 𝒫 ∪ ( M “ 𝐴 ) )
4	3	rexeqdv	⊢ ( 𝐴 ∈ On → ( ∃ 𝑟 ∈ 𝒫 ( O ‘ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ↔ ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
5	3 4	rexeqbidv	⊢ ( 𝐴 ∈ On → ( ∃ 𝑙 ∈ 𝒫 ( O ‘ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( O ‘ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
6	1 5	bitr4d	⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ( O ‘ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( O ‘ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )