Theorem opabbrex 5808

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 3985	. . 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 2225	. 2 |- ( ( V e. _V /\ E e. _V ) -> { z | E. f E. p ( z = <. f , p >. /\ th ) } e. _V )
4		df-opab 3985	. . 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 1854	. . . 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 3165	. . 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 3138	. 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 4063	1 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   _Vcvv 2681   <.cop 3525   class class class wbr 3924   {copab 3983  (class class class)co 5767

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-opab 3985

This theorem is referenced by: (None)