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Theorem opabex2 6719
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
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 opabex2.1 . . 3  |-  ( ph  ->  A  e.  V )
2 opabex2.2 . . 3  |-  ( ph  ->  B  e.  W )
3 xpexg 6709 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
41, 2, 3syl2anc 661 . 2  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
5 df-opab 4506 . . 3  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
6 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( z  =  <. x ,  y
>.  /\  ps ) )  ->  z  =  <. x ,  y >. )
7 opabex2.3 . . . . . . . . . 10  |-  ( (
ph  /\  ps )  ->  x  e.  A )
8 opabex2.4 . . . . . . . . . 10  |-  ( (
ph  /\  ps )  ->  y  e.  B )
9 opelxpi 5030 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( A  X.  B
) )
107, 8, 9syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ps )  -> 
<. x ,  y >.  e.  ( A  X.  B
) )
1110adantrl 715 . . . . . . . 8  |-  ( (
ph  /\  ( z  =  <. x ,  y
>.  /\  ps ) )  ->  <. x ,  y
>.  e.  ( A  X.  B ) )
126, 11eqeltrd 2555 . . . . . . 7  |-  ( (
ph  /\  ( z  =  <. x ,  y
>.  /\  ps ) )  ->  z  e.  ( A  X.  B ) )
1312ex 434 . . . . . 6  |-  ( ph  ->  ( ( z  = 
<. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B
) ) )
1413exlimdvv 1701 . . . . 5  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B
) ) )
1514alrimiv 1695 . . . 4  |-  ( ph  ->  A. z ( E. x E. y ( z  =  <. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B ) ) )
16 abss 3569 . . . 4  |-  ( { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) }  C_  ( A  X.  B
)  <->  A. z ( E. x E. y ( z  =  <. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B ) ) )
1715, 16sylibr 212 . . 3  |-  ( ph  ->  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) } 
C_  ( A  X.  B ) )
185, 17syl5eqss 3548 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  C_  ( A  X.  B
) )
194, 18ssexd 4594 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   _Vcvv 3113    C_ wss 3476   <.cop 4033   {copab 4504    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-opab 4506  df-xp 5005
This theorem is referenced by:  legval  23698
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