Theorem opabbrex 5491

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 3810	. . 3 |- { <.f , p >. | th } = {z | E.fE. p(z = <.f , p >. /\ th) }
2		opabbrex.2	. . 3 |- (( V e. _V /\ E e. _V) -> { <.f , p >. | th } e. _V)
3	1, 2	syl5eqelr 2122	. 2 |- (( V e. _V /\ E e. _V) -> {z | E.fE. p(z = <.f , p >. /\ th) } e. _V)
4		df-opab 3810	. . 3 |- { <.f , p >. | (f( V W E) p /\ ps) } = {z | E.fE. 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 264	. . . . . 6 |- (( V e. _V /\ E e. _V) -> ((f( V W E) p /\ ps) -> th))
7	6	anim2d 320	. . . . 5 |- (( V e. _V /\ E e. _V) -> ((z = <.f , p >. /\ (f( V W E) p /\ ps)) -> (z = <.f , p >. /\ th)))
8	7	2eximdv 1759	. . . 4 |- (( V e. _V /\ E e. _V) -> (E.fE. p(z = <.f , p >. /\ (f( V W E) p /\ ps)) -> E.fE. p(z = <.f , p >. /\ th)))
9	8	ss2abdv 3007	. . 3 |- (( V e. _V /\ E e. _V) -> {z | E.fE. p(z = <.f , p >. /\ (f( V W E) p /\ ps)) } C_ {z | E.fE. p(z = <.f , p >. /\ th) })
10	4, 9	syl5eqss 2983	. 2 |- (( V e. _V /\ E e. _V) -> { <.f , p >. | (f( V W E) p /\ ps) } C_ {z | E.fE. p(z = <.f , p >. /\ th) })
11	3, 10	ssexd 3888	1 |- (( V e. _V /\ E e. _V) -> { <.f , p >. | (f( V W E) p /\ ps) } e. _V)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023   _Vcvv 2551   <.cop 3370   class class class wbr 3755   {copab 3808  (class class class)co 5455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-opab 3810

This theorem is referenced by:  sprmpt2  5798