Theorem trpredex

Description: The transitive predecessors under a relation form a set.

This is the first theorem in the transitive predecessor series that requires the axiom of infinity. (Contributed by Scott Fenton, 18-Feb-2011)

	Ref	Expression
Assertion	trpredex	⊢ TrPred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V

Proof

Step	Hyp	Ref	Expression
1		df-trpred	⊢ TrPred ( 𝑅 , 𝐴 , 𝑋 ) = ∪ ran ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω )
2		frfnom	⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) Fn ω
3		omex	⊢ ω ∈ V
4		fnex	⊢ ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) Fn ω ∧ ω ∈ V ) → ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ∈ V )
5	2 3 4	mp2an	⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ∈ V
6	5	rnex	⊢ ran ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ∈ V
7	6	uniex	⊢ ∪ ran ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ∈ V
8	1 7	eqeltri	⊢ TrPred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V