Theorem opabbrex 6064

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 4151	. . 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 2319	. 2 |- ( ( V e. _V /\ E e. _V ) -> { z | E. f E. p ( z = <. f , p >. /\ th ) } e. _V )
4		df-opab 4151	. . 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 279	. . . . . 6 |- ( ( V e. _V /\ E e. _V ) -> ( ( f ( V W E ) p /\ ps ) -> th ) )
7	6	anim2d 337	. . . . 5 |- ( ( V e. _V /\ E e. _V ) -> ( ( z = <. f , p >. /\ ( f ( V W E ) p /\ ps ) ) -> ( z = <. f , p >. /\ th ) ) )
8	7	2eximdv 1930	. . . 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 3300	. . 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 3273	. 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 4229	1 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   _Vcvv 2802   <.cop 3672   class class class wbr 4088   {copab 4149  (class class class)co 6017

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-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-opab 4151

This theorem is referenced by: (None)