Theorem rnoprab 6284

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 6242	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	rneqi 5142	. 2 |- ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
3		rnopab 5160	. 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 1860	. . . 4 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) )
5		opex 4626	. . . . . . 7 |- <. x , y >. e. _V
6	5	isseti 3040	. . . . . 6 |- E. w w = <. x , y >.
7		19.41v 1779	. . . . . 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 1676	. . . 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 2516	. 2 |- { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph }
12	2, 3, 11	3eqtri 2415	1 |- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }

Syntax hints:   /\ wa 367   = wceq 1399   E.wex 1620   {cab 2367   <.cop 3950   {copab 4424   ran crn 4914   {coprab 6197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-cnv 4921  df-dm 4923  df-rn 4924  df-oprab 6200

This theorem is referenced by:  rnoprab2  6285  elrnmpt2res  6315  ellines  29955