Theorem wfr2a

Description: A weak version of wfr2 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020)

	Ref	Expression
Hypotheses	wfr2a.1	⊢ 𝑅 We 𝐴
	wfr2a.2	⊢ 𝑅 Se 𝐴
	wfr2a.3	⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion	wfr2a	⊢ ( 𝑋 ∈ dom 𝐹 → ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		wfr2a.1	⊢ 𝑅 We 𝐴
2		wfr2a.2	⊢ 𝑅 Se 𝐴
3		wfr2a.3	⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
4		fveq2	⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) )
5		predeq3	⊢ ( 𝑥 = 𝑋 → Pred ( 𝑅 , 𝐴 , 𝑥 ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) )
6	5	reseq2d	⊢ ( 𝑥 = 𝑋 → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑥 ) ) = ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) )
7	6	fveq2d	⊢ ( 𝑥 = 𝑋 → ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑥 ) ) ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )
8	4 7	eqeq12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑥 ) ) ) ↔ ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) ) )
9	1 2 3	wfrlem12	⊢ ( 𝑥 ∈ dom 𝐹 → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑥 ) ) ) )
10	8 9	vtoclga	⊢ ( 𝑋 ∈ dom 𝐹 → ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) )