Theorem rnoprab 5854

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

Assertion

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

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem rnoprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfoprab2 5818	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	rneqi 4767	. 2 |- ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
3		rnopab 4786	. 2 |- ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) }
4		exrot3 1668	. . . 4 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) )
5		vex 2689	. . . . . . . 8 |- x e. _V
6		vex 2689	. . . . . . . 8 |- y e. _V
7	5, 6	opex 4151	. . . . . . 7 |- <. x , y >. e. _V
8	7	isseti 2694	. . . . . 6 |- E. w w = <. x , y >.
9		19.41v 1874	. . . . . 6 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ( E. w w = <. x , y >. /\ ph ) )
10	8, 9	mpbiran 924	. . . . 5 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ph )
11	10	2exbii 1585	. . . 4 |- ( E. x E. y E. w ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
12	4, 11	bitri 183	. . 3 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
13	12	abbii 2255	. 2 |- { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph }
14	2, 3, 13	3eqtri 2164	1 |- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   {cab 2125   <.cop 3530   {copab 3988   ran crn 4540   {coprab 5775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550  df-oprab 5778

This theorem is referenced by:  rnoprab2  5855