Theorem opabssxpd 4788

Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 6390. (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, A   y, A   x, B   y, B   ph, x   ph, 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 4174	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
2		simprl 531	. . . . . 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 4784	. . . . . . 7 |- ( ( ph /\ ps ) -> <. x , y >. e. ( A X. B ) )
6	5	adantrl 478	. . . . . 6 |- ( ( ph /\ ( z = <. x , y >. /\ ps ) ) -> <. x , y >. e. ( A X. B ) )
7	2, 6	eqeltrd 2311	. . . . 5 |- ( ( ph /\ ( z = <. x , y >. /\ ps ) ) -> z e. ( A X. B ) )
8	7	ex 115	. . . 4 |- ( ph -> ( ( z = <. x , y >. /\ ps ) -> z e. ( A X. B ) ) )
9	8	exlimdvv 1949	. . 3 |- ( ph -> ( E. x E. y ( z = <. x , y >. /\ ps ) -> z e. ( A X. B ) ) )
10	9	abssdv 3314	. 2 |- ( ph -> { z | E. x E. y ( z = <. x , y >. /\ ps ) } C_ ( A X. B ) )
11	1, 10	eqsstrid 3286	1 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   C_ wss 3213   <.cop 3694   {copab 4172   X. cxp 4749

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-opab 4174  df-xp 4757

This theorem is referenced by:  opabex2  6390