Theorem opabbi2dv 4503

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2197. (Contributed by NM, 24-Feb-2014.)

Hypotheses

Ref	Expression
opabbi2dv.1	|- Rel A
opabbi2dv.3	|- ( ph -> ( <. x , y >. e. A <-> ps ) )

Assertion

Ref	Expression
opabbi2dv	|- ( ph -> A = { <. x , y >. | ps } )

Distinct variable groups:   x, y, A   ph, x, y

Allowed substitution hints:   ps( x, y)

Proof of Theorem opabbi2dv

Step	Hyp	Ref	Expression
1		opabbi2dv.1	. . 3 |- Rel A
2		opabid2 4485	. . 3 |- ( Rel A -> { <. x , y >. | <. x , y >. e. A } = A )
3	1, 2	ax-mp 7	. 2 |- { <. x , y >. | <. x , y >. e. A } = A
4		opabbi2dv.3	. . 3 |- ( ph -> ( <. x , y >. e. A <-> ps ) )
5	4	opabbidv 3844	. 2 |- ( ph -> { <. x , y >. | <. x , y >. e. A } = { <. x , y >. | ps } )
6	3, 5	syl5eqr 2127	1 |- ( ph -> A = { <. x , y >. | ps } )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   {copab 3838   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by: (None)