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Theorem dfco2 5188
Description: Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
Assertion
Ref Expression
dfco2  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem dfco2
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5187 . 2  |-  Rel  ( A  o.  B )
2 reliun 4822 . . 3  |-  ( Rel  U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  A. x  e.  _V  Rel  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
3 relxp 4810 . . . 4  |-  Rel  (
( `' B " { x } )  X.  ( A " { x } ) )
43a1i 10 . . 3  |-  ( x  e.  _V  ->  Rel  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
52, 4mprgbir 2626 . 2  |-  Rel  U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x }
) )
6 vex 2804 . . . 4  |-  y  e. 
_V
7 vex 2804 . . . 4  |-  z  e. 
_V
8 opelco2g 4867 . . . 4  |-  ( ( y  e.  _V  /\  z  e.  _V )  ->  ( <. y ,  z
>.  e.  ( A  o.  B )  <->  E. x
( <. y ,  x >.  e.  B  /\  <. x ,  z >.  e.  A
) ) )
96, 7, 8mp2an 653 . . 3  |-  ( <.
y ,  z >.  e.  ( A  o.  B
)  <->  E. x ( <.
y ,  x >.  e.  B  /\  <. x ,  z >.  e.  A
) )
10 eliun 3925 . . . 4  |-  ( <.
y ,  z >.  e.  U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x  e.  _V  <.
y ,  z >.  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
11 rexv 2815 . . . 4  |-  ( E. x  e.  _V  <. y ,  z >.  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x <. y ,  z >.  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
12 opelxp 4735 . . . . . 6  |-  ( <.
y ,  z >.  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  ( y  e.  ( `' B " { x } )  /\  z  e.  ( A " { x } ) ) )
13 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
1413, 6elimasn 5054 . . . . . . . 8  |-  ( y  e.  ( `' B " { x } )  <->  <. x ,  y >.  e.  `' B )
1513, 6opelcnv 4879 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  `' B  <->  <. y ,  x >.  e.  B )
1614, 15bitri 240 . . . . . . 7  |-  ( y  e.  ( `' B " { x } )  <->  <. y ,  x >.  e.  B )
1713, 7elimasn 5054 . . . . . . 7  |-  ( z  e.  ( A " { x } )  <->  <. x ,  z >.  e.  A )
1816, 17anbi12i 678 . . . . . 6  |-  ( ( y  e.  ( `' B " { x } )  /\  z  e.  ( A " {
x } ) )  <-> 
( <. y ,  x >.  e.  B  /\  <. x ,  z >.  e.  A
) )
1912, 18bitri 240 . . . . 5  |-  ( <.
y ,  z >.  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  ( <. y ,  x >.  e.  B  /\  <. x ,  z
>.  e.  A ) )
2019exbii 1572 . . . 4  |-  ( E. x <. y ,  z
>.  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  E. x
( <. y ,  x >.  e.  B  /\  <. x ,  z >.  e.  A
) )
2110, 11, 203bitrri 263 . . 3  |-  ( E. x ( <. y ,  x >.  e.  B  /\  <. x ,  z
>.  e.  A )  <->  <. y ,  z >.  e.  U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x }
) ) )
229, 21bitri 240 . 2  |-  ( <.
y ,  z >.  e.  ( A  o.  B
)  <->  <. y ,  z
>.  e.  U_ x  e. 
_V  ( ( `' B " { x } )  X.  ( A " { x }
) ) )
231, 5, 22eqrelriiv 4797 1  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   {csn 3653   <.cop 3656   U_ciun 3921    X. cxp 4703   `'ccnv 4704   "cima 4708    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  dfco2a  5189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iun 3923  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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