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Metamath Proof Explorer

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 ) )