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Theorem dfco2a 4899
Description: Generalization of dfco2 4898, 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 4898 . 2  |-  ( A  o.  B )  = 
U_ x  e.  _V  ( ( `' B " { x } )  X.  ( A " { x } ) )
2 vex 2618 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 vex 2618 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
43eliniseg 4771 . . . . . . . . . . . . . 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 4638 . . . . . . . . . . . . 13  |-  ( z B x  ->  x  e.  ran  B )
75, 6sylbi 119 . . . . . . . . . . . 12  |-  ( z  e.  ( `' B " { x } )  ->  x  e.  ran  B )
8 vex 2618 . . . . . . . . . . . . . 14  |-  w  e. 
_V
92, 8elimasn 4768 . . . . . . . . . . . . 13  |-  ( w  e.  ( A " { x } )  <->  <. x ,  w >.  e.  A )
102, 8opeldm 4609 . . . . . . . . . . . . 13  |-  ( <.
x ,  w >.  e.  A  ->  x  e.  dom  A )
119, 10sylbi 119 . . . . . . . . . . . 12  |-  ( w  e.  ( A " { x } )  ->  x  e.  dom  A )
127, 11anim12ci 332 . . . . . . . . . . 11  |-  ( ( z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) )  ->  ( x  e. 
dom  A  /\  x  e.  ran  B ) )
1312adantl 271 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  (
z  e.  ( `' B " { x } )  /\  w  e.  ( A " {
x } ) ) )  ->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1413exlimivv 1821 . . . . . . . . 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 4430 . . . . . . . . 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 3172 . . . . . . . . 9  |-  ( x  e.  ( dom  A  i^i  ran  B )  <->  ( x  e.  dom  A  /\  x  e.  ran  B ) )
1714, 15, 163imtr4i 199 . . . . . . . 8  |-  ( y  e.  ( ( `' B " { x } )  X.  ( A " { x }
) )  ->  x  e.  ( dom  A  i^i  ran 
B ) )
18 ssel 3008 . . . . . . . 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 386 . . . . . 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 1750 . . . . 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 2631 . . . . 5  |-  ( E. x  e.  _V  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) )  <->  E. x  y  e.  ( ( `' B " { x } )  X.  ( A " { x } ) ) )
23 df-rex 2361 . . . . 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 221 . . . 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 3719 . . . 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 3719 . . . 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 221 . . 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 2083 . 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 2129 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 102    <-> wb 103    = wceq 1287   E.wex 1424    e. wcel 1436   E.wrex 2356   _Vcvv 2615    i^i cin 2987    C_ wss 2988   {csn 3431   <.cop 3434   U_ciun 3715   class class class wbr 3822    X. cxp 4411   `'ccnv 4412   dom cdm 4413   ran crn 4414   "cima 4416    o. ccom 4417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-iun 3717  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426
This theorem is referenced by: (None)
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