Theorem elopab 4709

Description: Membership in a class abstraction of 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 3040	. 2 |- ( A e. { <. x , y >. | ph } -> A e. _V )
2		opex 4664	. . . . 5 |- <. x , y >. e. _V
3		eleq1 2537	. . . . 5 |- ( A = <. x , y >. -> ( A e. _V <-> <. x , y >. e. _V ) )
4	2, 3	mpbiri 241	. . . 4 |- ( A = <. x , y >. -> A e. _V )
5	4	adantr 472	. . 3 |- ( ( A = <. x , y >. /\ ph ) -> A e. _V )
6	5	exlimivv 1786	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
7		eqeq1 2475	. . . . 5 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
8	7	anbi1d 719	. . . 4 |- ( z = A -> ( ( z = <. x , y >. /\ ph ) <-> ( A = <. x , y >. /\ ph ) ) )
9	8	2exbidv 1778	. . 3 |- ( z = A -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
10		df-opab 4455	. . 3 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
11	9, 10	elab2g 3175	. 2 |- ( A e. _V -> ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
12	1, 6, 11	pm5.21nii 360	1 |- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   <.cop 3965   {copab 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455

This theorem is referenced by:  opelopabsbALT  4710  opelopabsb  4711  opelopabt  4713  opelopabga  4714  opabn0  4732  iunopab  4737  elopabr  4738  epelg  4751  elxp  4856  elopaelxp  4912  elopaba  4952  elcnv  5016  dfmpt3  5708  fmptsng  6101  fmptsnd  6102  0neqopab  6353  opabex3d  6790  opabex3  6791  fsplit  6920  rtrclreclem3  13200  isfunc  15847  usgraop  25156  clwlkswlks  25565  brabgaf  28292  qqhval2  28860  eulerpartlemgvv  29282  poimirlem26  32030  dicelval3  34819  pellexlem5  35748  pellex  35750  opelopab4  36988  griedg0ssusgr  39501  rgrusgrprc  39793