Theorem opabex2 7227

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		xpexg 6960	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> ( A X. B ) e. _V )
5		opabex2.3	. . 3 |- ( ( ph /\ ps ) -> x e. A )
6		opabex2.4	. . 3 |- ( ( ph /\ ps ) -> y e. B )
7	5, 6	opabssxpd 5338	. 2 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )
8	4, 7	ssexd 4805	1 |- ( ph -> { <. x , y >. | ps } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   {copab 4712   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  legval  25479  wksv  26515  rfovcnvfvd  38301  sprsymrelfvlem  41740