Metamath Proof Explorer


Theorem elold

Description: Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024)

Ref Expression
Assertion elold ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) ↔ ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) )

Proof

Step Hyp Ref Expression
1 oldval ( 𝐴 ∈ On → ( O ‘ 𝐴 ) = ( M “ 𝐴 ) )
2 1 eleq2d ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) ↔ 𝑋 ( M “ 𝐴 ) ) )
3 eluni ( 𝑋 ( M “ 𝐴 ) ↔ ∃ 𝑦 ( 𝑋𝑦𝑦 ∈ ( M “ 𝐴 ) ) )
4 madef M : On ⟶ 𝒫 No
5 ffn ( M : On ⟶ 𝒫 No → M Fn On )
6 4 5 ax-mp M Fn On
7 onss ( 𝐴 ∈ On → 𝐴 ⊆ On )
8 fvelimab ( ( M Fn On ∧ 𝐴 ⊆ On ) → ( 𝑦 ∈ ( M “ 𝐴 ) ↔ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) )
9 6 7 8 sylancr ( 𝐴 ∈ On → ( 𝑦 ∈ ( M “ 𝐴 ) ↔ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) )
10 9 anbi2d ( 𝐴 ∈ On → ( ( 𝑋𝑦𝑦 ∈ ( M “ 𝐴 ) ) ↔ ( 𝑋𝑦 ∧ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) )
11 10 exbidv ( 𝐴 ∈ On → ( ∃ 𝑦 ( 𝑋𝑦𝑦 ∈ ( M “ 𝐴 ) ) ↔ ∃ 𝑦 ( 𝑋𝑦 ∧ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) )
12 fvex ( M ‘ 𝑏 ) ∈ V
13 12 clel3 ( 𝑋 ∈ ( M ‘ 𝑏 ) ↔ ∃ 𝑦 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) )
14 13 rexbii ( ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ↔ ∃ 𝑏𝐴𝑦 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) )
15 rexcom4 ( ∃ 𝑏𝐴𝑦 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) ↔ ∃ 𝑦𝑏𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) )
16 eqcom ( 𝑦 = ( M ‘ 𝑏 ) ↔ ( M ‘ 𝑏 ) = 𝑦 )
17 16 anbi2ci ( ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) ↔ ( 𝑋𝑦 ∧ ( M ‘ 𝑏 ) = 𝑦 ) )
18 17 rexbii ( ∃ 𝑏𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) ↔ ∃ 𝑏𝐴 ( 𝑋𝑦 ∧ ( M ‘ 𝑏 ) = 𝑦 ) )
19 r19.42v ( ∃ 𝑏𝐴 ( 𝑋𝑦 ∧ ( M ‘ 𝑏 ) = 𝑦 ) ↔ ( 𝑋𝑦 ∧ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) )
20 18 19 bitri ( ∃ 𝑏𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) ↔ ( 𝑋𝑦 ∧ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) )
21 20 exbii ( ∃ 𝑦𝑏𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋𝑦 ) ↔ ∃ 𝑦 ( 𝑋𝑦 ∧ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) )
22 14 15 21 3bitrri ( ∃ 𝑦 ( 𝑋𝑦 ∧ ∃ 𝑏𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ↔ ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) )
23 11 22 bitrdi ( 𝐴 ∈ On → ( ∃ 𝑦 ( 𝑋𝑦𝑦 ∈ ( M “ 𝐴 ) ) ↔ ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) )
24 3 23 syl5bb ( 𝐴 ∈ On → ( 𝑋 ( M “ 𝐴 ) ↔ ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) )
25 2 24 bitrd ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) ↔ ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) )