Metamath Proof Explorer

Theorem fpr2

Description: Law of founded partial recursion, part two. Now we establish the value of F within A . (Contributed by Scott Fenton, 11-Sep-2023)

Ref Expression
Hypothesis fprr.1 ⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion fpr2 ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )

Proof

Step Hyp Ref Expression
1 fprr.1 ⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
2 1 fpr1 ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )
3 2 fndmd ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = 𝐴 )
4 3 eleq2d ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴 ) )
5 4 biimpar ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ dom 𝐹 )
6 fveq2 ⊢ ( 𝑦 = 𝑋 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑋 ) )
7 id ⊢ ( 𝑦 = 𝑋 → 𝑦 = 𝑋 )
8 predeq3 ⊢ ( 𝑦 = 𝑋 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) )
9 8 reseq2d ⊢ ( 𝑦 = 𝑋 → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) )
10 7 9 oveq12d ⊢ ( 𝑦 = 𝑋 → ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )
11 6 10 eqeq12d ⊢ ( 𝑦 = 𝑋 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) )
12 11 imbi2d ⊢ ( 𝑦 = 𝑋 → ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) ) )
13 eqid ⊢ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) }
14 13 frrlem1 ⊢ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
15 14 1 fprlem1 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑔 ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ∧ ℎ ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ) ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) )
16 14 1 15 frrlem10 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑦 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
17 16 expcom ⊢ ( 𝑦 ∈ dom 𝐹 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) )
18 12 17 vtoclga ⊢ ( 𝑋 ∈ dom 𝐹 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) )
19 18 impcom ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )
20 5 19 syldan ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )