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Metamath Proof Explorer

Theorem r1val2

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of Monk1 p. 113. (Contributed by NM, 30-Nov-2003)

		Ref	Expression
	Assertion	r1val2	⊢ ( 𝐴 ∈ On → ( 𝑅₁ ‘ 𝐴 ) = { 𝑥 ∣ ( rank ‘ 𝑥 ) ∈ 𝐴 } )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	rankr1a	⊢ ( 𝐴 ∈ On → ( 𝑥 ∈ ( 𝑅₁ ‘ 𝐴 ) ↔ ( rank ‘ 𝑥 ) ∈ 𝐴 ) )
3	2	eqabdv	⊢ ( 𝐴 ∈ On → ( 𝑅₁ ‘ 𝐴 ) = { 𝑥 ∣ ( rank ‘ 𝑥 ) ∈ 𝐴 } )