Theorem rnoprab 6171

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 6135	. . 3 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	rneqi 5010	. 2 |- ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
3		rnopab 5029	. 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 1742	. . . 4 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) )
5		vex 2824	. . . . . . . 8 |- x e. _V
6		vex 2824	. . . . . . . 8 |- y e. _V
7	5, 6	opex 4369	. . . . . . 7 |- <. x , y >. e. _V
8	7	isseti 2830	. . . . . 6 |- E. w w = <. x , y >.
9		19.41v 1958	. . . . . 6 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ( E. w w = <. x , y >. /\ ph ) )
10	8, 9	mpbiran 953	. . . . 5 |- ( E. w ( w = <. x , y >. /\ ph ) <-> ph )
11	10	2exbii 1659	. . . 4 |- ( E. x E. y E. w ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
12	4, 11	bitri 184	. . 3 |- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ph )
13	12	abbii 2354	. 2 |- { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph }
14	2, 3, 13	3eqtri 2263	1 |- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   E.wex 1545   {cab 2224   <.cop 3712   {copab 4191   ran crn 4775   {coprab 6086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785  df-oprab 6089

This theorem is used by:  rnoprab2  6172