Theorem opabssxpd 5338

Description: An ordered-pair class abstraction is a subset of an Cartesian product. Formerly part of proof for opabex2 7227. (Contributed by AV, 26-Nov-2021.)

Hypotheses

Ref	Expression
opabssxpd.x	|- ( ( ph /\ ps ) -> x e. A )
opabssxpd.y	|- ( ( ph /\ ps ) -> y e. B )

Assertion

Ref	Expression
opabssxpd	|- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )

Distinct variable groups:   x, y, A   x, B, y   ph, x, y

Allowed substitution hints:   ps( x, y)

Proof of Theorem opabssxpd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 4713	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
2		simprl 794	. . . . . 6 |- ( ( ph /\ ( z = <. x , y >. /\ ps ) ) -> z = <. x , y >. )
3		opabssxpd.x	. . . . . . . 8 |- ( ( ph /\ ps ) -> x e. A )
4		opabssxpd.y	. . . . . . . 8 |- ( ( ph /\ ps ) -> y e. B )
5	3, 4	opelxpd 5149	. . . . . . 7 |- ( ( ph /\ ps ) -> <. x , y >. e. ( A X. B ) )
6	5	adantrl 752	. . . . . 6 |- ( ( ph /\ ( z = <. x , y >. /\ ps ) ) -> <. x , y >. e. ( A X. B ) )
7	2, 6	eqeltrd 2701	. . . . 5 |- ( ( ph /\ ( z = <. x , y >. /\ ps ) ) -> z e. ( A X. B ) )
8	7	ex 450	. . . 4 |- ( ph -> ( ( z = <. x , y >. /\ ps ) -> z e. ( A X. B ) ) )
9	8	exlimdvv 1862	. . 3 |- ( ph -> ( E. x E. y ( z = <. x , y >. /\ ps ) -> z e. ( A X. B ) ) )
10	9	abssdv 3676	. 2 |- ( ph -> { z | E. x E. y ( z = <. x , y >. /\ ps ) } C_ ( A X. B ) )
11	1, 10	syl5eqss 3649	1 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   C_ wss 3574   <.cop 4183   {copab 4712   X. cxp 5112

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  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by:  opabex2  7227  uspgropssxp  41752