Theorem opabbi2dv 4811

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2312. (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 4793	. . 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 4095	. 2 |- ( ph -> { <. x , y >. | <. x , y >. e. A } = { <. x , y >. | ps } )
6	3, 5	eqtr3id 2240	1 |- ( ph -> A = { <. x , y >. | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   <.cop 3621   {copab 4089   Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665  df-rel 4666

This theorem is referenced by: (None)