Theorem elopab 4236

Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
elopab	|- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem elopab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2737	. 2 |- ( A e. { <. x , y >. | ph } -> A e. _V )
2		vex 2729	. . . . . 6 |- x e. _V
3		vex 2729	. . . . . 6 |- y e. _V
4	2, 3	opex 4207	. . . . 5 |- <. x , y >. e. _V
5		eleq1 2229	. . . . 5 |- ( A = <. x , y >. -> ( A e. _V <-> <. x , y >. e. _V ) )
6	4, 5	mpbiri 167	. . . 4 |- ( A = <. x , y >. -> A e. _V )
7	6	adantr 274	. . 3 |- ( ( A = <. x , y >. /\ ph ) -> A e. _V )
8	7	exlimivv 1884	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
9		eqeq1 2172	. . . . 5 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
10	9	anbi1d 461	. . . 4 |- ( z = A -> ( ( z = <. x , y >. /\ ph ) <-> ( A = <. x , y >. /\ ph ) ) )
11	10	2exbidv 1856	. . 3 |- ( z = A -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
12		df-opab 4044	. . 3 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
13	11, 12	elab2g 2873	. 2 |- ( A e. _V -> ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
14	1, 8, 13	pm5.21nii 694	1 |- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   <.cop 3579   {copab 4042

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044

This theorem is referenced by:  opelopabsbALT  4237  opelopabsb  4238  opelopabt  4240  opelopabga  4241  opabm  4258  iunopab  4259  epelg  4268  elxp  4621  elco  4770  elcnv  4781  dfmpt3  5310  0neqopab  5887  brabvv  5888  opabex3d  6089  opabex3  6090