Theorem rdgsucg

Description: The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014)

	Ref	Expression
Assertion	rdgsucg	⊢ ( 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝐵 ) = ( 𝐹 ‘ ( rec ( 𝐹 , 𝐴 ) ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rdgdmlim	⊢ Lim dom rec ( 𝐹 , 𝐴 )
2		limsuc	⊢ ( Lim dom rec ( 𝐹 , 𝐴 ) → ( 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) ↔ suc 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) ↔ suc 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) )
4		eqid	⊢ ( 𝑥 ∈ V ↦ if ( 𝑥 = ∅ , 𝐴 , if ( Lim dom 𝑥 , ∪ ran 𝑥 , ( 𝐹 ‘ ( 𝑥 ‘ ∪ dom 𝑥 ) ) ) ) ) = ( 𝑥 ∈ V ↦ if ( 𝑥 = ∅ , 𝐴 , if ( Lim dom 𝑥 , ∪ ran 𝑥 , ( 𝐹 ‘ ( 𝑥 ‘ ∪ dom 𝑥 ) ) ) ) )
5		rdgvalg	⊢ ( 𝑦 ∈ dom rec ( 𝐹 , 𝐴 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( ( 𝑥 ∈ V ↦ if ( 𝑥 = ∅ , 𝐴 , if ( Lim dom 𝑥 , ∪ ran 𝑥 , ( 𝐹 ‘ ( 𝑥 ‘ ∪ dom 𝑥 ) ) ) ) ) ‘ ( rec ( 𝐹 , 𝐴 ) ↾ 𝑦 ) ) )
6		rdgseg	⊢ ( 𝑦 ∈ dom rec ( 𝐹 , 𝐴 ) → ( rec ( 𝐹 , 𝐴 ) ↾ 𝑦 ) ∈ V )
7		rdgfun	⊢ Fun rec ( 𝐹 , 𝐴 )
8		funfn	⊢ ( Fun rec ( 𝐹 , 𝐴 ) ↔ rec ( 𝐹 , 𝐴 ) Fn dom rec ( 𝐹 , 𝐴 ) )
9	7 8	mpbi	⊢ rec ( 𝐹 , 𝐴 ) Fn dom rec ( 𝐹 , 𝐴 )
10		limord	⊢ ( Lim dom rec ( 𝐹 , 𝐴 ) → Ord dom rec ( 𝐹 , 𝐴 ) )
11	1 10	ax-mp	⊢ Ord dom rec ( 𝐹 , 𝐴 )
12	4 5 6 9 11	tz7.44-2	⊢ ( suc 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝐵 ) = ( 𝐹 ‘ ( rec ( 𝐹 , 𝐴 ) ‘ 𝐵 ) ) )
13	3 12	sylbi	⊢ ( 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝐵 ) = ( 𝐹 ‘ ( rec ( 𝐹 , 𝐴 ) ‘ 𝐵 ) ) )