Theorem elopab 4376

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 2825	. 2 |- ( A e. { <. x , y >. | ph } -> A e. _V )
2		vex 2816	. . . . . 6 |- x e. _V
3		vex 2816	. . . . . 6 |- y e. _V
4	2, 3	opex 4345	. . . . 5 |- <. x , y >. e. _V
5		eleq1 2295	. . . . 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 1946	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
9		eqeq1 2239	. . . . 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 1917	. . 3 |- ( z = A -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
12		df-opab 4172	. . 3 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
13	11, 12	elab2g 2964	. 2 |- ( A e. _V -> ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
14	1, 8, 13	pm5.21nii 712	1 |- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   <.cop 3692   {copab 4170

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172

This theorem is referenced by:  opelopabsbALT  4377  opelopabsb  4378  opelopabt  4380  opelopabga  4381  opabm  4399  iunopab  4400  epelg  4411  elxp  4766  elco  4921  elcnv  4932  dfmpt3  5481  0neqopab  6098  brabvv  6099  opabex3d  6314  opabex3  6315  griedg0ssusgr  16246