Theorem opabssxpd 4762

Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 6357. (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 4151	. 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 4758	. . . . . . 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 2308	. . . . 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 1946	. . 3 |- ( ph -> ( E. x E. y ( z = <. x , y >. /\ ps ) -> z e. ( A X. B ) ) )
10	9	abssdv 3301	. 2 |- ( ph -> { z | E. x E. y ( z = <. x , y >. /\ ps ) } C_ ( A X. B ) )
11	1, 10	eqsstrid 3273	1 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   C_ wss 3200   <.cop 3672   {copab 4149   X. cxp 4723

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-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731

This theorem is referenced by:  opabex2  6357