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Theorem dfco2a 5173
Description: Generalization of dfco2 5172, 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 5172 . 2  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
2 vex 2791 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 vex 2791 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
43eliniseg 5042 . . . . . . . . . . . . . 14  |-  ( x  e.  _V  ->  (
z  e.  ( `' B " { x } )  <->  z B x ) )
52, 4ax-mp 8 . . . . . . . . . . . . 13  |-  ( z  e.  ( `' B " { x } )  <-> 
z B x )
63, 2brelrn 4909 . . . . . . . . . . . . 13  |-  ( z B x  ->  x  e.  ran  B )
75, 6sylbi 187 . . . . . . . . . . . 12  |-  ( z  e.  ( `' B " { x } )  ->  x  e.  ran  B )
8 vex 2791 . . . . . . . . . . . . . 14  |-  w  e. 
_V
92, 8elimasn 5038 . . . . . . . . . . . . 13  |-  ( w  e.  ( A " { x } )  <->  <. x ,  w >.  e.  A )
102, 8opeldm 4882 . . . . . . . . . . . . 13  |-  ( <.
x ,  w >.  e.  A  ->  x  e.  dom  A )
119, 10sylbi 187 . . . . . . . . . . . 12  |-  ( w  e.  ( A " { x } )  ->  x  e.  dom  A )
127, 11anim12ci 550 . . . . . . . . . . 11  |-  ( ( z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) )  ->  ( x  e. 
dom  A  /\  x  e.  ran  B ) )
1312adantl 452 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  (
z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) )  ->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1413exlimivv 1667 . . . . . . . . 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 4706 . . . . . . . . 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 3358 . . . . . . . . 9  |-  ( x  e.  ( dom  A  i^i  ran  B )  <->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1714, 15, 163imtr4i 257 . . . . . . . 8  |-  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  ( dom  A  i^i  ran 
B ) )
18 ssel 3174 . . . . . . . 8  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( x  e.  ( dom 
A  i^i  ran  B )  ->  x  e.  C
) )
1917, 18syl5 28 . . . . . . 7  |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  C ) )
2019pm4.71rd 616 . . . . . 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 1612 . . . . 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 2802 . . . . 5  |-  ( E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
23 df-rex 2549 . . . . 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 279 . . . 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 3909 . . . 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 3909 . . . 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 279 . . 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 2281 . 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 2327 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    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693
This theorem is referenced by:  fparlem3  6220  fparlem4  6221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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