Metamath Proof Explorer


Theorem elmade

Description: Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024)

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

Proof

Step Hyp Ref Expression
1 madef M : On ⟶ 𝒫 No
2 1 ffvelrni ( 𝐴 ∈ On → ( M ‘ 𝐴 ) ∈ 𝒫 No )
3 2 elpwid ( 𝐴 ∈ On → ( M ‘ 𝐴 ) ⊆ No )
4 3 sseld ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) → 𝑋 No ) )
5 scutcl ( 𝑙 <<s 𝑟 → ( 𝑙 |s 𝑟 ) ∈ No )
6 eleq1 ( ( 𝑙 |s 𝑟 ) = 𝑋 → ( ( 𝑙 |s 𝑟 ) ∈ No 𝑋 No ) )
7 6 biimpd ( ( 𝑙 |s 𝑟 ) = 𝑋 → ( ( 𝑙 |s 𝑟 ) ∈ No 𝑋 No ) )
8 5 7 mpan9 ( ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 No )
9 8 rexlimivw ( ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 No )
10 9 rexlimivw ( ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 No )
11 10 a1i ( 𝐴 ∈ On → ( ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 No ) )
12 madeval2 ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = { 𝑥 No ∣ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) } )
13 12 eleq2d ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ 𝑋 ∈ { 𝑥 No ∣ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) } ) )
14 eqeq2 ( 𝑥 = 𝑋 → ( ( 𝑙 |s 𝑟 ) = 𝑥 ↔ ( 𝑙 |s 𝑟 ) = 𝑋 ) )
15 14 anbi2d ( 𝑥 = 𝑋 → ( ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) ↔ ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
16 15 2rexbidv ( 𝑥 = 𝑋 → ( ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) ↔ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
17 16 elrab3 ( 𝑋 No → ( 𝑋 ∈ { 𝑥 No ∣ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) } ↔ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
18 13 17 sylan9bb ( ( 𝐴 ∈ On ∧ 𝑋 No ) → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )
19 18 ex ( 𝐴 ∈ On → ( 𝑋 No → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) )
20 4 11 19 pm5.21ndd ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) )