Theorem opabex2 6352

Description: Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)

Hypotheses

Ref	Expression
opabex2.1	|- ( ph -> A e. V )
opabex2.2	|- ( ph -> B e. W )
opabex2.3	|- ( ( ph /\ ps ) -> x e. A )
opabex2.4	|- ( ( ph /\ ps ) -> y e. B )

Assertion

Ref	Expression
opabex2	|- ( ph -> { <. x , y >. | ps } e. _V )

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

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

Proof of Theorem opabex2

Step	Hyp	Ref	Expression
1		opabex2.1	. . 3 |- ( ph -> A e. V )
2		opabex2.2	. . 3 |- ( ph -> B e. W )
3	1, 2	xpexd 4839	. 2 |- ( ph -> ( A X. B ) e. _V )
4		opabex2.3	. . 3 |- ( ( ph /\ ps ) -> x e. A )
5		opabex2.4	. . 3 |- ( ( ph /\ ps ) -> y e. B )
6	4, 5	opabssxpd 4760	. 2 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )
7	3, 6	ssexd 4227	1 |- ( ph -> { <. x , y >. | ps } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   _Vcvv 2800   {copab 4147   X. cxp 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-opab 4149  df-xp 4729

This theorem is referenced by: (None)