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Theorem dfco2a 4764
Description: Generalization of dfco2 4763, where  C can have any value between  dom  i^i  ran and  _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfco2a  dom  i^i  ran  C_  C  o.  U_  C  `' " { }  X.  " { }
Distinct variable groups:   ,   ,   , C

Proof of Theorem dfco2a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfco2 4763 . 2  o. 
U_  _V  `' " { }  X.  " { }
2 vex 2554 . . . . . . . . . . . . . 14 
_V
3 vex 2554 . . . . . . . . . . . . . . 15 
_V
43eliniseg 4638 . . . . . . . . . . . . . 14  _V  `' " { }
52, 4ax-mp 7 . . . . . . . . . . . . 13  `' " { }
63, 2brelrn 4510 . . . . . . . . . . . . 13  ran
75, 6sylbi 114 . . . . . . . . . . . 12  `' " { }  ran
8 vex 2554 . . . . . . . . . . . . . 14 
_V
92, 8elimasn 4635 . . . . . . . . . . . . 13  " { }  <. ,  >.
102, 8opeldm 4481 . . . . . . . . . . . . 13  <. ,  >.  dom
119, 10sylbi 114 . . . . . . . . . . . 12  " { }  dom
127, 11anim12ci 322 . . . . . . . . . . 11  `' " { }  " { }  dom  ran
1312adantl 262 . . . . . . . . . 10  <. ,  >.  `' " { }  " { }  dom  ran
1413exlimivv 1773 . . . . . . . . 9  <. ,  >.  `' " { }  " { }  dom  ran
15 elxp 4305 . . . . . . . . 9  `' " { }  X.  " { } 
<. ,  >.  `' " { }  " { }
16 elin 3120 . . . . . . . . 9  dom  i^i  ran  dom  ran
1714, 15, 163imtr4i 190 . . . . . . . 8  `' " { }  X.  " { }  dom  i^i  ran
18 ssel 2933 . . . . . . . 8  dom  i^i  ran  C_  C  dom  i^i  ran  C
1917, 18syl5 28 . . . . . . 7  dom  i^i  ran  C_  C  `' " { }  X.  " { }  C
2019pm4.71rd 374 . . . . . 6  dom  i^i  ran  C_  C  `' " { }  X.  " { }  C  `' " { }  X.  " { }
2120exbidv 1703 . . . . 5  dom  i^i  ran  C_  C  `' " { }  X.  " { }  C  `' " { }  X.  " { }
22 rexv 2566 . . . . 5  _V  `' " { }  X.  " { }  `' " { }  X.  " { }
23 df-rex 2306 . . . . 5  C  `' " { }  X.  " { }  C  `' " { }  X.  " { }
2421, 22, 233bitr4g 212 . . . 4  dom  i^i  ran  C_  C  _V  `' " { }  X.  " { }  C  `' " { }  X.  " { }
25 eliun 3652 . . . 4  U_ 
_V  `' " { }  X.  " { }  _V  `' " { }  X.  " { }
26 eliun 3652 . . . 4  U_  C  `' " { }  X.  " { }  C  `' " { }  X.  " { }
2724, 25, 263bitr4g 212 . . 3  dom  i^i  ran  C_  C  U_  _V  `' " { }  X.  " { }  U_  C  `' " { }  X.  " { }
2827eqrdv 2035 . 2  dom  i^i  ran  C_  C  U_  _V  `' " { }  X.  " { }  U_  C  `' " { }  X.  " { }
291, 28syl5eq 2081 1  dom  i^i  ran  C_  C  o.  U_  C  `' " { }  X.  " { }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wrex 2301   _Vcvv 2551    i^i cin 2910    C_ wss 2911   {csn 3367   <.cop 3370   U_ciun 3648   class class class wbr 3755    X. cxp 4286   `'ccnv 4287   dom cdm 4288   ran crn 4289   "cima 4291    o. ccom 4292
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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