Metamath Proof Explorer


Theorem frrlem7

Description: Lemma for well-founded recursion. The well-founded recursion generator's domain is a subclass of A . (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 frrlem7 dom 𝐹𝐴

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 1 2 frrlem5 𝐹 = 𝐵
4 3 dmeqi dom 𝐹 = dom 𝐵
5 dmuni dom 𝐵 = 𝑔𝐵 dom 𝑔
6 4 5 eqtri dom 𝐹 = 𝑔𝐵 dom 𝑔
7 6 sseq1i ( dom 𝐹𝐴 𝑔𝐵 dom 𝑔𝐴 )
8 iunss ( 𝑔𝐵 dom 𝑔𝐴 ↔ ∀ 𝑔𝐵 dom 𝑔𝐴 )
9 7 8 bitri ( dom 𝐹𝐴 ↔ ∀ 𝑔𝐵 dom 𝑔𝐴 )
10 1 frrlem3 ( 𝑔𝐵 → dom 𝑔𝐴 )
11 9 10 mprgbir dom 𝐹𝐴