Theorem opabresex2d 6696

Description: Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)

Hypotheses

Ref	Expression
opabresex2d.1	|- ( ( ph /\ x ( W ` G ) y ) -> ps )
opabresex2d.2	|- ( ph -> { <. x , y >. | ps } e. V )

Assertion

Ref	Expression
opabresex2d	|- ( ph -> { <. x , y >. | ( x ( W ` G ) y /\ th ) } e. _V )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)   th( x, y)   G( x, y)   V( x, y)   W( x, y)

Proof of Theorem opabresex2d

Step	Hyp	Ref	Expression
1		opabresex2d.1	. . . 4 |- ( ( ph /\ x ( W ` G ) y ) -> ps )
2	1	ex 450	. . 3 |- ( ph -> ( x ( W ` G ) y -> ps ) )
3	2	alrimivv 1856	. 2 |- ( ph -> A. x A. y ( x ( W ` G ) y -> ps ) )
4		opabresex2d.2	. 2 |- ( ph -> { <. x , y >. | ps } e. V )
5		opabbrex 6695	. 2 |- ( ( A. x A. y ( x ( W ` G ) y -> ps ) /\ { <. x , y >. | ps } e. V ) -> { <. x , y >. | ( x ( W ` G ) y /\ th ) } e. _V )
6	3, 4, 5	syl2anc 693	1 |- ( ph -> { <. x , y >. | ( x ( W ` G ) y /\ th ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-opab 4713

This theorem is referenced by:  mptmpt2opabbrd  7248