Metamath Proof Explorer


Theorem frrlem5

Description: Lemma for well-founded recursion. State the well-founded recursion generator in terms of the acceptable functions. (Contributed by Scott Fenton, 27-Aug-2022)

Ref Expression
Hypotheses frrlem5.1 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
frrlem5.2 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion frrlem5 𝐹 = 𝐵

Proof

Step Hyp Ref Expression
1 frrlem5.1 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
2 frrlem5.2 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
3 df-frecs frecs ( 𝑅 , 𝐴 , 𝐺 ) = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
4 1 unieqi 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
5 3 2 4 3eqtr4i 𝐹 = 𝐵