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Theorem opabex2 6619
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 6610 . . 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 4452 . . 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 4972 . . . . . . . . . 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 2539 . . . . . . 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 1692 . . . . 5  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B
) ) )
1514alrimiv 1686 . . . 4  |-  ( ph  ->  A. z ( E. x E. y ( z  =  <. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B ) ) )
16 abss 3522 . . . 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 3501 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  C_  ( A  X.  B
) )
194, 18ssexd 4540 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   _Vcvv 3071    C_ wss 3429   <.cop 3984   {copab 4450    X. cxp 4939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-opab 4452  df-xp 4947
This theorem is referenced by:  legval  23146
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