Theorem rdgeq2

Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995) (Revised by Mario Carneiro, 9-May-2015)

	Ref	Expression
Assertion	rdgeq2	⊢ ( 𝐴 = 𝐵 → rec ( 𝐹 , 𝐴 ) = rec ( 𝐹 , 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ifeq1	⊢ ( 𝐴 = 𝐵 → if ( 𝑔 = ∅ , 𝐴 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) = if ( 𝑔 = ∅ , 𝐵 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) )
2	1	mpteq2dv	⊢ ( 𝐴 = 𝐵 → ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐴 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) = ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐵 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) )
3		recseq	⊢ ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐴 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) = ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐵 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) → recs ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐴 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) ) = recs ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐵 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) ) )
4	2 3	syl	⊢ ( 𝐴 = 𝐵 → recs ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐴 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) ) = recs ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐵 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) ) )
5		df-rdg	⊢ rec ( 𝐹 , 𝐴 ) = recs ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐴 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) )
6		df-rdg	⊢ rec ( 𝐹 , 𝐵 ) = recs ( ( 𝑔 ∈ V ↦ if ( 𝑔 = ∅ , 𝐵 , if ( Lim dom 𝑔 , ∪ ran 𝑔 , ( 𝐹 ‘ ( 𝑔 ‘ ∪ dom 𝑔 ) ) ) ) ) )
7	4 5 6	3eqtr4g	⊢ ( 𝐴 = 𝐵 → rec ( 𝐹 , 𝐴 ) = rec ( 𝐹 , 𝐵 ) )