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Metamath Proof Explorer

Theorem rnoprab

Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004) (Revised by David Abernethy, 19-Apr-2013)

Ref Expression
Assertion rnoprab 
|- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }

Proof

Step Hyp Ref Expression
1 dfoprab2 
 |-  { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2 1 rneqi 
 |-  ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
3 rnopab 
 |-  ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) }
4 exrot3 
 |-  ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) )
5 opex 
 |-  <. x , y >. e. _V
6 5 isseti 
 |-  E. w w = <. x , y >.
7 19.41v 
 |-  ( E. w ( w = <. x , y >. /\ ph ) <-> ( E. w w = <. x , y >. /\ ph ) )
8 6 7 mpbiran 
 |-  ( E. w ( w = <. x , y >. /\ ph ) <-> ph )
9 8 2exbii 
 |-  ( E. x E. y E. w ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
10 4 9 bitri 
 |-  ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
11 10 abbii 
 |-  { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph }
12 2 3 11 3eqtri 
 |-  ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }