Theorem opabbi2dv 4815

Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2315. (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 4797	. . 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 4099	. 2 |- ( ph -> { <. x , y >. | <. x , y >. e. A } = { <. x , y >. | ps } )
6	3, 5	eqtr3id 2243	1 |- ( ph -> A = { <. x , y >. | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   <.cop 3625   {copab 4093   Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669  df-rel 4670

This theorem is referenced by: (None)