Theorem opabbrex 5897

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 4051	. . 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	eqeltrrid 2258	. 2 |- ( ( V e. _V /\ E e. _V ) -> { z | E. f E. p ( z = <. f , p >. /\ th ) } e. _V )
4		df-opab 4051	. . 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 277	. . . . . 6 |- ( ( V e. _V /\ E e. _V ) -> ( ( f ( V W E ) p /\ ps ) -> th ) )
7	6	anim2d 335	. . . . 5 |- ( ( V e. _V /\ E e. _V ) -> ( ( z = <. f , p >. /\ ( f ( V W E ) p /\ ps ) ) -> ( z = <. f , p >. /\ th ) ) )
8	7	2eximdv 1875	. . . 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 3220	. . 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	eqsstrid 3193	. 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 4129	1 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   _Vcvv 2730   <.cop 3586   class class class wbr 3989   {copab 4049  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-opab 4051

This theorem is referenced by: (None)