Theorem opabex2 6387

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 4864	. 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 4785	. 2 |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )
7	3, 6	ssexd 4249	1 |- ( ph -> { <. x , y >. | ps } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   _Vcvv 2812   {copab 4169   X. cxp 4746

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-opab 4171  df-xp 4754

This theorem is referenced by: (None)