Theorem rnoprab 6366

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 6324	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	rneqi 5215	. 2 |- ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
3		rnopab 5233	. 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 1836	. . . 4 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) )
5		opex 4697	. . . . . . 7 |- <. x , y >. e. _V
6	5	isseti 3099	. . . . . 6 |- E. w w = <. x , y >.
7		19.41v 1755	. . . . . 6 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ( E. w w = <. x , y >. /\ ph ) )
8	6, 7	mpbiran 916	. . . . 5 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ph )
9	8	2exbii 1653	. . . 4 |- ( E. x E. y E. w ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
10	4, 9	bitri 249	. . 3 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
11	10	abbii 2575	. 2 |- { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph }
12	2, 3, 11	3eqtri 2474	1 |- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }

Syntax hints:   /\ wa 369   = wceq 1381   E.wex 1597   {cab 2426   <.cop 4016   {copab 4490   ran crn 4986   {coprab 6278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-cnv 4993  df-dm 4995  df-rn 4996  df-oprab 6281

This theorem is referenced by:  rnoprab2  6367  elrnmpt2res  6397  ellines  29770