Theorem r1val3

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of Monk1 p. 113. (Contributed by NM, 30-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	r1val3	⊢ ( 𝐴 ∈ On → ( 𝑅1 ‘ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 𝒫 { 𝑦 ∣ ( rank ‘ 𝑦 ) ∈ 𝑥 } )

Proof

Step	Hyp	Ref	Expression
1		r1fnon	⊢ 𝑅1 Fn On
2	1	fndmi	⊢ dom 𝑅1 = On
3	2	eleq2i	⊢ ( 𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On )
4		r1val1	⊢ ( 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 𝒫 ( 𝑅1 ‘ 𝑥 ) )
5	3 4	sylbir	⊢ ( 𝐴 ∈ On → ( 𝑅1 ‘ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 𝒫 ( 𝑅1 ‘ 𝑥 ) )
6		onelon	⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ On )
7		r1val2	⊢ ( 𝑥 ∈ On → ( 𝑅1 ‘ 𝑥 ) = { 𝑦 ∣ ( rank ‘ 𝑦 ) ∈ 𝑥 } )
8	6 7	syl	⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ 𝐴 ) → ( 𝑅1 ‘ 𝑥 ) = { 𝑦 ∣ ( rank ‘ 𝑦 ) ∈ 𝑥 } )
9	8	pweqd	⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ 𝐴 ) → 𝒫 ( 𝑅1 ‘ 𝑥 ) = 𝒫 { 𝑦 ∣ ( rank ‘ 𝑦 ) ∈ 𝑥 } )
10	9	iuneq2dv	⊢ ( 𝐴 ∈ On → ∪ 𝑥 ∈ 𝐴 𝒫 ( 𝑅1 ‘ 𝑥 ) = ∪ 𝑥 ∈ 𝐴 𝒫 { 𝑦 ∣ ( rank ‘ 𝑦 ) ∈ 𝑥 } )
11	5 10	eqtrd	⊢ ( 𝐴 ∈ On → ( 𝑅1 ‘ 𝐴 ) = ∪ 𝑥 ∈ 𝐴 𝒫 { 𝑦 ∣ ( rank ‘ 𝑦 ) ∈ 𝑥 } )