Theorem opabssxpd 4806

Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 6418. (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 4188	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
2		simprl 535	. . . . . 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 4802	. . . . . . 7 |- ( ( ph /\ ps ) -> <. x , y >. e. ( A X. B ) )
6	5	adantrl 482	. . . . . 6 |- ( ( ph /\ ( z = <. x , y >. /\ ps ) ) -> <. x , y >. e. ( A X. B ) )
7	2, 6	eqeltrd 2315	. . . . 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 1953	. . 3 |- ( ph -> ( E. x E. y ( z = <. x , y >. /\ ps ) -> z e. ( A X. B ) ) )
10	9	abssdv 3322	. 2 |- ( ph -> { z | E. x E. y ( z = <. x , y >. /\ ps ) } C_ ( A X. B ) )
11	1, 10	eqsstrid 3294	1 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   C_ wss 3220   <.cop 3708   {copab 4186   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775

This theorem is referenced by:  opabex2  6418