Theorem rankvalg

Description: Value of the rank function. Definition 9.14 of TakeutiZaring p. 79 (proved as a theorem from our definition). This variant of rankval expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003)

	Ref	Expression
Assertion	rankvalg	⊢ ( 𝐴 ∈ 𝑉 → ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) } )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝑦 = 𝐴 → ( rank ‘ 𝑦 ) = ( rank ‘ 𝐴 ) )
2		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ ( 𝑅1 ‘ suc 𝑥 ) ↔ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) ) )
3	2	rabbidv	⊢ ( 𝑦 = 𝐴 → { 𝑥 ∈ On ∣ 𝑦 ∈ ( 𝑅1 ‘ suc 𝑥 ) } = { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) } )
4	3	inteqd	⊢ ( 𝑦 = 𝐴 → ∩ { 𝑥 ∈ On ∣ 𝑦 ∈ ( 𝑅1 ‘ suc 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) } )
5	1 4	eqeq12d	⊢ ( 𝑦 = 𝐴 → ( ( rank ‘ 𝑦 ) = ∩ { 𝑥 ∈ On ∣ 𝑦 ∈ ( 𝑅1 ‘ suc 𝑥 ) } ↔ ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) } ) )
6		vex	⊢ 𝑦 ∈ V
7	6	rankval	⊢ ( rank ‘ 𝑦 ) = ∩ { 𝑥 ∈ On ∣ 𝑦 ∈ ( 𝑅1 ‘ suc 𝑥 ) }
8	5 7	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) } )