Theorem rnoprab 6406

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 6364	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	rneqi 5080	. 2 |- ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
3		rnopab 5098	. 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 1942	. . . 4 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) )
5		opex 4678	. . . . . . 7 |- <. x , y >. e. _V
6	5	isseti 3063	. . . . . 6 |- E. w w = <. x , y >.
7		19.41v 1841	. . . . . 6 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ( E. w w = <. x , y >. /\ ph ) )
8	6, 7	mpbiran 934	. . . . 5 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ph )
9	8	2exbii 1730	. . . 4 |- ( E. x E. y E. w ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
10	4, 9	bitri 257	. . 3 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
11	10	abbii 2578	. 2 |- { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph }
12	2, 3, 11	3eqtri 2488	1 |- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }

Syntax hints:   /\ wa 375   = wceq 1455   E.wex 1674   {cab 2448   <.cop 3986   {copab 4474   ran crn 4854   {coprab 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-cnv 4861  df-dm 4863  df-rn 4864  df-oprab 6319

This theorem is referenced by:  rnoprab2  6407  elrnmpt2res  6437  ellines  30968