Theorem opabbi2dv 4777

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2296. (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 4759	. . 3 |- ( Rel A -> { <. x , y >. | <. x , y >. e. A } = A )
3	1, 2	ax-mp 5	. 2 |- { <. x , y >. | <. x , y >. e. A } = A
4		opabbi2dv.3	. . 3 |- ( ph -> ( <. x , y >. e. A <-> ps ) )
5	4	opabbidv 4070	. 2 |- ( ph -> { <. x , y >. | <. x , y >. e. A } = { <. x , y >. | ps } )
6	3, 5	eqtr3id 2224	1 |- ( ph -> A = { <. x , y >. | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   <.cop 3596   {copab 4064   Rel wrel 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-xp 4633  df-rel 4634

This theorem is referenced by: (None)