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Theorem dfco2a 5130
Description: Generalization of dfco2 5129, 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 5129 . 2  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
2 vex 2741 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 vex 2741 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
43eliniseg 4999 . . . . . . . . . . . . . 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 4861 . . . . . . . . . . . . 13  |-  ( z B x  ->  x  e.  ran  B )
75, 6sylbi 121 . . . . . . . . . . . 12  |-  ( z  e.  ( `' B " { x } )  ->  x  e.  ran  B )
8 vex 2741 . . . . . . . . . . . . . 14  |-  w  e. 
_V
92, 8elimasn 4996 . . . . . . . . . . . . 13  |-  ( w  e.  ( A " { x } )  <->  <. x ,  w >.  e.  A )
102, 8opeldm 4831 . . . . . . . . . . . . 13  |-  ( <.
x ,  w >.  e.  A  ->  x  e.  dom  A )
119, 10sylbi 121 . . . . . . . . . . . 12  |-  ( w  e.  ( A " { x } )  ->  x  e.  dom  A )
127, 11anim12ci 339 . . . . . . . . . . 11  |-  ( ( z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) )  ->  ( x  e. 
dom  A  /\  x  e.  ran  B ) )
1312adantl 277 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  (
z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) )  ->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1413exlimivv 1896 . . . . . . . . 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 4644 . . . . . . . . 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 3319 . . . . . . . . 9  |-  ( x  e.  ( dom  A  i^i  ran  B )  <->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1714, 15, 163imtr4i 201 . . . . . . . 8  |-  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  ( dom  A  i^i  ran 
B ) )
18 ssel 3150 . . . . . . . 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 394 . . . . . 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 1825 . . . . 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 2756 . . . . 5  |-  ( E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
23 df-rex 2461 . . . . 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 223 . . . 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 3891 . . . 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 3891 . . . 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 223 . . 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 2175 . 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, 28eqtrid 2222 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 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   _Vcvv 2738    i^i cin 3129    C_ wss 3130   {csn 3593   <.cop 3596   U_ciun 3887   class class class wbr 4004    X. cxp 4625   `'ccnv 4626   dom cdm 4627   ran crn 4628   "cima 4630    o. ccom 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-iun 3889  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640
This theorem is referenced by: (None)
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