Theorem jech9.3

Description: Every set belongs to some value of the cumulative hierarchy of sets function R1 , i.e. the indexed union of all values of R1 is the universe. Lemma 9.3 of Jech p. 71. (Contributed by NM, 4-Oct-2003) (Revised by Mario Carneiro, 8-Jun-2013)

	Ref	Expression
Assertion	jech9.3	⊢ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = V

Proof

Step	Hyp	Ref	Expression
1		r1fnon	⊢ 𝑅1 Fn On
2		fniunfv	⊢ ( 𝑅1 Fn On → ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ran 𝑅1 )
3	1 2	ax-mp	⊢ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ran 𝑅1
4		fndm	⊢ ( 𝑅1 Fn On → dom 𝑅1 = On )
5	1 4	ax-mp	⊢ dom 𝑅1 = On
6	5	imaeq2i	⊢ ( 𝑅1 “ dom 𝑅1 ) = ( 𝑅1 “ On )
7		imadmrn	⊢ ( 𝑅1 “ dom 𝑅1 ) = ran 𝑅1
8	6 7	eqtr3i	⊢ ( 𝑅1 “ On ) = ran 𝑅1
9	8	unieqi	⊢ ∪ ( 𝑅1 “ On ) = ∪ ran 𝑅1
10		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
11	3 9 10	3eqtr2i	⊢ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = V