Theorem elopab 4346

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 2811	. 2 |- ( A e. { <. x , y >. | ph } -> A e. _V )
2		vex 2802	. . . . . 6 |- x e. _V
3		vex 2802	. . . . . 6 |- y e. _V
4	2, 3	opex 4315	. . . . 5 |- <. x , y >. e. _V
5		eleq1 2292	. . . . 5 |- ( A = <. x , y >. -> ( A e. _V <-> <. x , y >. e. _V ) )
6	4, 5	mpbiri 168	. . . 4 |- ( A = <. x , y >. -> A e. _V )
7	6	adantr 276	. . 3 |- ( ( A = <. x , y >. /\ ph ) -> A e. _V )
8	7	exlimivv 1943	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
9		eqeq1 2236	. . . . 5 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
10	9	anbi1d 465	. . . 4 |- ( z = A -> ( ( z = <. x , y >. /\ ph ) <-> ( A = <. x , y >. /\ ph ) ) )
11	10	2exbidv 1914	. . 3 |- ( z = A -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
12		df-opab 4146	. . 3 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
13	11, 12	elab2g 2950	. 2 |- ( A e. _V -> ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
14	1, 8, 13	pm5.21nii 709	1 |- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   <.cop 3669   {copab 4144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146

This theorem is referenced by:  opelopabsbALT  4347  opelopabsb  4348  opelopabt  4350  opelopabga  4351  opabm  4369  iunopab  4370  epelg  4381  elxp  4736  elco  4888  elcnv  4899  dfmpt3  5446  0neqopab  6049  brabvv  6050  opabex3d  6266  opabex3  6267