Metamath Proof Explorer

Theorem fpr3g

Description: Functions defined by well-founded recursion over a partial order are identical up to relation, domain, and characteristic function. This version of frr3g does not require infinity. (Contributed by Scott Fenton, 24-Aug-2022)

Ref Expression
Assertion fpr3g ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐹 = 𝐺 )

Proof

Step Hyp Ref Expression
1 eqidd ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐴 = 𝐴 )
2 r19.21v ⊢ ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) ↔ ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
3 simprll ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ) → 𝐹 Fn 𝐴 )
4 simprrl ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ) → 𝐺 Fn 𝐴 )
5 predss ⊢ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴
6 fvreseq ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴 ) → ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ↔ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
7 5 6 mpan2 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ↔ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
8 3 4 7 syl2anc ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ) → ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ↔ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
9 8 biimp3ar ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
10 9 oveq2d ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
11 fveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) )
12 id ⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 )
13 predeq3 ⊢ ( 𝑦 = 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑧 ) )
14 13 reseq2d ⊢ ( 𝑦 = 𝑧 → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
15 12 14 oveq12d ⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
16 11 15 eqeq12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) )
17 simp2lr ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
18 simp1 ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → 𝑧 ∈ 𝐴 )
19 16 17 18 rspcdva ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
20 fveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) )
21 13 reseq2d ⊢ ( 𝑦 = 𝑧 → ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
22 12 21 oveq12d ⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
23 20 22 eqeq12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) )
24 simp2rr ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
25 23 24 18 rspcdva ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
26 10 19 25 3eqtr4d ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) )
27 26 3exp ⊢ ( 𝑧 ∈ 𝐴 → ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
28 27 a2d ⊢ ( 𝑧 ∈ 𝐴 → ( ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
29 2 28 syl5bi ⊢ ( 𝑧 ∈ 𝐴 → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
30 fveq2 ⊢ ( 𝑧 = 𝑤 → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑤 ) )
31 fveq2 ⊢ ( 𝑧 = 𝑤 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑤 ) )
32 30 31 eqeq12d ⊢ ( 𝑧 = 𝑤 → ( ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ↔ ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
33 32 imbi2d ⊢ ( 𝑧 = 𝑤 → ( ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ↔ ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) ) )
34 29 33 frpoins2g ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑧 ∈ 𝐴 ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
35 r19.21v ⊢ ( ∀ 𝑧 ∈ 𝐴 ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ↔ ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
36 34 35 sylib ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
37 36 3impib ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) )
38 simp2l ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐹 Fn 𝐴 )
39 simp3l ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐺 Fn 𝐴 )
40 eqfnfv2 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ( 𝐴 = 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
41 38 39 40 syl2anc ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 = 𝐺 ↔ ( 𝐴 = 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
42 1 37 41 mpbir2and ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐹 = 𝐺 )