Theorem opabbrex 5569

Description: A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)

Hypotheses

Ref	Expression
opabbrex.1	|- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) )
opabbrex.2	|- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | th } e. _V )

Assertion

Ref	Expression
opabbrex	|- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )

Distinct variable groups:   f, E, p   f, V, p

Allowed substitution hints:   ps( f, p)   th( f, p)   W( f, p)

Proof of Theorem opabbrex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 3840	. . 3 |- { <. f , p >. | th } = { z | E. f E. p ( z = <. f , p >. /\ th ) }
2		opabbrex.2	. . 3 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | th } e. _V )
3	1, 2	syl5eqelr 2166	. 2 |- ( ( V e. _V /\ E e. _V ) -> { z | E. f E. p ( z = <. f , p >. /\ th ) } e. _V )
4		df-opab 3840	. . 3 |- { <. f , p >. | ( f ( V W E ) p /\ ps ) } = { z | E. f E. p ( z = <. f , p >. /\ ( f ( V W E ) p /\ ps ) ) }
5		opabbrex.1	. . . . . . 7 |- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) )
6	5	adantrd 273	. . . . . 6 |- ( ( V e. _V /\ E e. _V ) -> ( ( f ( V W E ) p /\ ps ) -> th ) )
7	6	anim2d 330	. . . . 5 |- ( ( V e. _V /\ E e. _V ) -> ( ( z = <. f , p >. /\ ( f ( V W E ) p /\ ps ) ) -> ( z = <. f , p >. /\ th ) ) )
8	7	2eximdv 1803	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( E. f E. p ( z = <. f , p >. /\ ( f ( V W E ) p /\ ps ) ) -> E. f E. p ( z = <. f , p >. /\ th ) ) )
9	8	ss2abdv 3067	. . 3 |- ( ( V e. _V /\ E e. _V ) -> { z | E. f E. p ( z = <. f , p >. /\ ( f ( V W E ) p /\ ps ) ) } C_ { z | E. f E. p ( z = <. f , p >. /\ th ) } )
10	4, 9	syl5eqss 3043	. 2 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } C_ { z | E. f E. p ( z = <. f , p >. /\ th ) } )
11	3, 10	ssexd 3918	1 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   _Vcvv 2601   <.cop 3401   class class class wbr 3785   {copab 3838  (class class class)co 5532

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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-opab 3840

This theorem is referenced by:  sprmpt2  5880