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Theorem dfco2a 4821
Description: Generalization of dfco2 4820, 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 4820 . 2  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
2 vex 2560 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 vex 2560 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
43eliniseg 4695 . . . . . . . . . . . . . 14  |-  ( x  e.  _V  ->  (
z  e.  ( `' B " { x } )  <->  z B x ) )
52, 4ax-mp 7 . . . . . . . . . . . . 13  |-  ( z  e.  ( `' B " { x } )  <-> 
z B x )
63, 2brelrn 4567 . . . . . . . . . . . . 13  |-  ( z B x  ->  x  e.  ran  B )
75, 6sylbi 114 . . . . . . . . . . . 12  |-  ( z  e.  ( `' B " { x } )  ->  x  e.  ran  B )
8 vex 2560 . . . . . . . . . . . . . 14  |-  w  e. 
_V
92, 8elimasn 4692 . . . . . . . . . . . . 13  |-  ( w  e.  ( A " { x } )  <->  <. x ,  w >.  e.  A )
102, 8opeldm 4538 . . . . . . . . . . . . 13  |-  ( <.
x ,  w >.  e.  A  ->  x  e.  dom  A )
119, 10sylbi 114 . . . . . . . . . . . 12  |-  ( w  e.  ( A " { x } )  ->  x  e.  dom  A )
127, 11anim12ci 322 . . . . . . . . . . 11  |-  ( ( z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) )  ->  ( x  e. 
dom  A  /\  x  e.  ran  B ) )
1312adantl 262 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  (
z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) )  ->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1413exlimivv 1776 . . . . . . . . 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 4362 . . . . . . . . 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 3126 . . . . . . . . 9  |-  ( x  e.  ( dom  A  i^i  ran  B )  <->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1714, 15, 163imtr4i 190 . . . . . . . 8  |-  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  ( dom  A  i^i  ran 
B ) )
18 ssel 2939 . . . . . . . 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 374 . . . . . 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 1706 . . . . 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 2572 . . . . 5  |-  ( E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
23 df-rex 2312 . . . . 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 212 . . . 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 3661 . . . 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 3661 . . . 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 212 . . 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 2038 . 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 2084 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 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   _Vcvv 2557    i^i cin 2916    C_ wss 2917   {csn 3375   <.cop 3378   U_ciun 3657   class class class wbr 3764    X. cxp 4343   `'ccnv 4344   dom cdm 4345   ran crn 4346   "cima 4348    o. ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by: (None)
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