Theorem rdglimg

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

	Ref	Expression
Assertion	rdglimg	⊢ ( ( 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) ∧ Lim 𝐵 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝐵 ) = ∪ ( rec ( 𝐹 , 𝐴 ) “ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		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 ) ) ) ) )
2		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 ) ) )
3		rdgseg	⊢ ( 𝑦 ∈ dom rec ( 𝐹 , 𝐴 ) → ( rec ( 𝐹 , 𝐴 ) ↾ 𝑦 ) ∈ V )
4		rdgfun	⊢ Fun rec ( 𝐹 , 𝐴 )
5		funfn	⊢ ( Fun rec ( 𝐹 , 𝐴 ) ↔ rec ( 𝐹 , 𝐴 ) Fn dom rec ( 𝐹 , 𝐴 ) )
6	4 5	mpbi	⊢ rec ( 𝐹 , 𝐴 ) Fn dom rec ( 𝐹 , 𝐴 )
7		rdgdmlim	⊢ Lim dom rec ( 𝐹 , 𝐴 )
8		limord	⊢ ( Lim dom rec ( 𝐹 , 𝐴 ) → Ord dom rec ( 𝐹 , 𝐴 ) )
9	7 8	ax-mp	⊢ Ord dom rec ( 𝐹 , 𝐴 )
10	1 2 3 6 9	tz7.44-3	⊢ ( ( 𝐵 ∈ dom rec ( 𝐹 , 𝐴 ) ∧ Lim 𝐵 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝐵 ) = ∪ ( rec ( 𝐹 , 𝐴 ) “ 𝐵 ) )