Theorem fpr2a

Description: Weak version of fpr2 which is useful for proofs that avoid the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024)

	Ref	Expression
Hypothesis	fpr2a.1	⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion	fpr2a	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		fpr2a.1	⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
2		fveq2	⊢ ( 𝑦 = 𝑋 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑋 ) )
3		id	⊢ ( 𝑦 = 𝑋 → 𝑦 = 𝑋 )
4		predeq3	⊢ ( 𝑦 = 𝑋 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) )
5	4	reseq2d	⊢ ( 𝑦 = 𝑋 → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) )
6	3 5	oveq12d	⊢ ( 𝑦 = 𝑋 → ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )
7	2 6	eqeq12d	⊢ ( 𝑦 = 𝑋 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) )
8	7	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 ) ) ) ) ) )
9		eqid	⊢ { 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 ) ) ) ) } = { 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 ) ) ) ) }
10	9 1	fprlem1	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑔 ∈ { 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 ) ) ) ) } ∧ ℎ ∈ { 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 ) ) ) ) } ) ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) )
11	9 1 10	frrlem10	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑦 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
12	11	expcom	⊢ ( 𝑦 ∈ dom 𝐹 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) )
13	8 12	vtoclga	⊢ ( 𝑋 ∈ dom 𝐹 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) )
14	13	impcom	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑋 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )