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Theorem opabex2 6711
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 6575 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
41, 2, 3syl2anc 659 . 2  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
5 df-opab 4498 . . 3  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
6 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( z  =  <. x ,  y
>.  /\  ps ) )  ->  z  =  <. x ,  y >. )
7 opabex2.3 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  x  e.  A )
8 opabex2.4 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  y  e.  B )
9 opelxpi 5020 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( A  X.  B
) )
107, 8, 9syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  ps )  -> 
<. x ,  y >.  e.  ( A  X.  B
) )
1110adantrl 713 . . . . . . 7  |-  ( (
ph  /\  ( z  =  <. x ,  y
>.  /\  ps ) )  ->  <. x ,  y
>.  e.  ( A  X.  B ) )
126, 11eqeltrd 2542 . . . . . 6  |-  ( (
ph  /\  ( z  =  <. x ,  y
>.  /\  ps ) )  ->  z  e.  ( A  X.  B ) )
1312ex 432 . . . . 5  |-  ( ph  ->  ( ( z  = 
<. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B
) ) )
1413exlimdvv 1730 . . . 4  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  ->  z  e.  ( A  X.  B
) ) )
1514abssdv 3560 . . 3  |-  ( ph  ->  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) } 
C_  ( A  X.  B ) )
165, 15syl5eqss 3533 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  C_  ( A  X.  B
) )
174, 16ssexd 4584 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   _Vcvv 3106   <.cop 4022   {copab 4496    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-opab 4498  df-xp 4994
This theorem is referenced by:  legval  24175
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