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Theorem dfco2a 5046
Description: Generalization of dfco2 5045, where  C can have any value between  dom  A  i^i  ran 
B and  _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfco2a  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( A  o.  B )  =  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem dfco2a
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfco2 5045 . 2  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
2 vex 2692 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 vex 2692 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
43eliniseg 4916 . . . . . . . . . . . . . 14  |-  ( x  e.  _V  ->  (
z  e.  ( `' B " { x } )  <->  z B x ) )
52, 4ax-mp 5 . . . . . . . . . . . . 13  |-  ( z  e.  ( `' B " { x } )  <-> 
z B x )
63, 2brelrn 4779 . . . . . . . . . . . . 13  |-  ( z B x  ->  x  e.  ran  B )
75, 6sylbi 120 . . . . . . . . . . . 12  |-  ( z  e.  ( `' B " { x } )  ->  x  e.  ran  B )
8 vex 2692 . . . . . . . . . . . . . 14  |-  w  e. 
_V
92, 8elimasn 4913 . . . . . . . . . . . . 13  |-  ( w  e.  ( A " { x } )  <->  <. x ,  w >.  e.  A )
102, 8opeldm 4749 . . . . . . . . . . . . 13  |-  ( <.
x ,  w >.  e.  A  ->  x  e.  dom  A )
119, 10sylbi 120 . . . . . . . . . . . 12  |-  ( w  e.  ( A " { x } )  ->  x  e.  dom  A )
127, 11anim12ci 337 . . . . . . . . . . 11  |-  ( ( z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) )  ->  ( x  e. 
dom  A  /\  x  e.  ran  B ) )
1312adantl 275 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  (
z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) )  ->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1413exlimivv 1869 . . . . . . . . 9  |-  ( E. z E. w ( y  =  <. z ,  w >.  /\  (
z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) )  ->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
15 elxp 4563 . . . . . . . . 9  |-  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  E. z E. w ( y  = 
<. z ,  w >.  /\  ( z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) ) )
16 elin 3263 . . . . . . . . 9  |-  ( x  e.  ( dom  A  i^i  ran  B )  <->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1714, 15, 163imtr4i 200 . . . . . . . 8  |-  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  ( dom  A  i^i  ran 
B ) )
18 ssel 3095 . . . . . . . 8  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( x  e.  ( dom 
A  i^i  ran  B )  ->  x  e.  C
) )
1917, 18syl5 32 . . . . . . 7  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  C ) )
2019pm4.71rd 392 . . . . . 6  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  ( x  e.  C  /\  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) ) ) )
2120exbidv 1798 . . . . 5  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( E. x  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x ( x  e.  C  /\  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) ) ) )
22 rexv 2707 . . . . 5  |-  ( E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
23 df-rex 2423 . . . . 5  |-  ( E. x  e.  C  y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  E. x
( x  e.  C  /\  y  e.  (
( `' B " { x } )  X.  ( A " { x } ) ) ) )
2421, 22, 233bitr4g 222 . . . 4  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  E. x  e.  C  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) ) )
25 eliun 3824 . . . 4  |-  ( y  e.  U_ x  e. 
_V  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
26 eliun 3824 . . . 4  |-  ( y  e.  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  E. x  e.  C  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
2724, 25, 263bitr4g 222 . . 3  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( y  e.  U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x }
) )  <->  y  e.  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x } ) ) ) )
2827eqrdv 2138 . 2  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  U_ x  e.  _V  (
( `' B " { x } )  X.  ( A " { x } ) )  =  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
) ) )
291, 28syl5eq 2185 1  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( A  o.  B )  =  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481   E.wrex 2418   _Vcvv 2689    i^i cin 3074    C_ wss 3075   {csn 3531   <.cop 3534   U_ciun 3820   class class class wbr 3936    X. cxp 4544   `'ccnv 4545   dom cdm 4546   ran crn 4547   "cima 4549    o. ccom 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-iun 3822  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559
This theorem is referenced by: (None)
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