Theorem opabbi2dv 4879

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2350. (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 4861	. . 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 4155	. 2 |- ( ph -> { <. x , y >. | <. x , y >. e. A } = { <. x , y >. | ps } )
6	3, 5	eqtr3id 2278	1 |- ( ph -> A = { <. x , y >. | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   <.cop 3672   {copab 4149   Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-rel 4732

This theorem is referenced by: (None)