This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )