Theorem opabex2 6362

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 4843	. 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 4764	. 2 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )
7	3, 6	ssexd 4230	1 |- ( ph -> { <. x , y >. | ps } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2201   _Vcvv 2801   {copab 4150   X. cxp 4725

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-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-opab 4152  df-xp 4733

This theorem is referenced by: (None)