Metamath Proof Explorer


Theorem madeval2

Description: Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021)

Ref Expression
Assertion madeval2 ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = { 𝑥 No ∣ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } )

Proof

Step Hyp Ref Expression
1 madeval ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) )
2 scutcl ( 𝑎 <<s 𝑏 → ( 𝑎 |s 𝑏 ) ∈ No )
3 eleq1 ( ( 𝑎 |s 𝑏 ) = 𝑥 → ( ( 𝑎 |s 𝑏 ) ∈ No 𝑥 No ) )
4 3 biimpd ( ( 𝑎 |s 𝑏 ) = 𝑥 → ( ( 𝑎 |s 𝑏 ) ∈ No 𝑥 No ) )
5 2 4 mpan9 ( ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) → 𝑥 No )
6 5 rexlimivw ( ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) → 𝑥 No )
7 6 rexlimivw ( ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) → 𝑥 No )
8 7 pm4.71ri ( ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( 𝑥 No ∧ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) )
9 8 abbii { 𝑥 ∣ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } = { 𝑥 ∣ ( 𝑥 No ∧ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) }
10 eleq1 ( 𝑦 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝑦 ∈ <<s ↔ ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ) )
11 breq1 ( 𝑦 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝑦 |s 𝑥 ↔ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
12 10 11 anbi12d ( 𝑦 = ⟨ 𝑎 , 𝑏 ⟩ → ( ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) ) )
13 12 rexxp ( ∃ 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ↔ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
14 imaindm ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) = ( |s “ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ dom |s ) )
15 dmscut dom |s = <<s
16 15 ineq2i ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ dom |s ) = ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s )
17 16 imaeq2i ( |s “ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ dom |s ) ) = ( |s “ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) )
18 14 17 eqtri ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) = ( |s “ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) )
19 18 eleq2i ( 𝑥 ∈ ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) ↔ 𝑥 ∈ ( |s “ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) ) )
20 vex 𝑥 ∈ V
21 20 elima ( 𝑥 ∈ ( |s “ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) ) ↔ ∃ 𝑦 ∈ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) 𝑦 |s 𝑥 )
22 elin ( 𝑦 ∈ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) ↔ ( 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∧ 𝑦 ∈ <<s ) )
23 22 anbi1i ( ( 𝑦 ∈ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) ∧ 𝑦 |s 𝑥 ) ↔ ( ( 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥 ) )
24 anass ( ( ( 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥 ) ↔ ( 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∧ ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ) )
25 23 24 bitri ( ( 𝑦 ∈ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) ∧ 𝑦 |s 𝑥 ) ↔ ( 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∧ ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ) )
26 25 rexbii2 ( ∃ 𝑦 ∈ ( ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ∩ <<s ) 𝑦 |s 𝑥 ↔ ∃ 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) )
27 19 21 26 3bitri ( 𝑥 ∈ ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) ↔ ∃ 𝑦 ∈ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) )
28 df-br ( 𝑎 <<s 𝑏 ↔ ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s )
29 28 anbi1i ( ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) )
30 df-ov ( 𝑎 |s 𝑏 ) = ( |s ‘ ⟨ 𝑎 , 𝑏 ⟩ )
31 30 eqeq1i ( ( 𝑎 |s 𝑏 ) = 𝑥 ↔ ( |s ‘ ⟨ 𝑎 , 𝑏 ⟩ ) = 𝑥 )
32 scutf |s : <<s ⟶ No
33 ffn ( |s : <<s ⟶ No → |s Fn <<s )
34 32 33 ax-mp |s Fn <<s
35 fnbrfvb ( ( |s Fn <<s ∧ ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ) → ( ( |s ‘ ⟨ 𝑎 , 𝑏 ⟩ ) = 𝑥 ↔ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
36 34 35 mpan ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s → ( ( |s ‘ ⟨ 𝑎 , 𝑏 ⟩ ) = 𝑥 ↔ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
37 31 36 syl5bb ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s → ( ( 𝑎 |s 𝑏 ) = 𝑥 ↔ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
38 37 pm5.32i ( ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
39 29 38 bitri ( ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
40 39 2rexbii ( ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( ⟨ 𝑎 , 𝑏 ⟩ ∈ <<s ∧ ⟨ 𝑎 , 𝑏 ⟩ |s 𝑥 ) )
41 13 27 40 3bitr4i ( 𝑥 ∈ ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) ↔ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) )
42 41 abbi2i ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) = { 𝑥 ∣ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) }
43 df-rab { 𝑥 No ∣ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } = { 𝑥 ∣ ( 𝑥 No ∧ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) }
44 9 42 43 3eqtr4i ( |s “ ( 𝒫 ( M “ 𝐴 ) × 𝒫 ( M “ 𝐴 ) ) ) = { 𝑥 No ∣ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) }
45 1 44 eqtrdi ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = { 𝑥 No ∣ ∃ 𝑎 ∈ 𝒫 ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } )