Theorem elopab 4381

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 2827	. 2 |- ( A e. { <. x , y >. | ph } -> A e. _V )
2		vex 2818	. . . . . 6 |- x e. _V
3		vex 2818	. . . . . 6 |- y e. _V
4	2, 3	opex 4350	. . . . 5 |- <. x , y >. e. _V
5		eleq1 2297	. . . . 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 1948	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
9		eqeq1 2241	. . . . 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 4177	. . 3 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
13	11, 12	elab2g 2967	. 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 2205   _Vcvv 2815   <.cop 3697   {copab 4175

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177

This theorem is referenced by:  opelopabsbALT  4382  opelopabsb  4383  opelopabt  4385  opelopabga  4386  opabm  4404  iunopab  4405  epelg  4416  elxp  4771  elco  4926  elcnv  4937  dfmpt3  5486  0neqopab  6106  brabvv  6107  opabex3d  6323  opabex3  6324  griedg0ssusgr  16372