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Theorem preoran2 25230
Description: The range of a preset equals its field. (Contributed by FL, 22-May-2011.)
Hypothesis
Ref Expression
preoref12.1  |-  X  =  dom  R
Assertion
Ref Expression
preoran2  |-  ( R  e. PresetRel  ->  ran  R  =  X )

Proof of Theorem preoran2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isprsr 25224 . . 3  |-  ( R  e. PresetRel  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
2 idref 5759 . . . . 5  |-  ( (  _I  |`  U. U. R
)  C_  R  <->  A. x  e.  U. U. R x R x )
3 preoref12.1 . . . . . 6  |-  X  =  dom  R
4 domrngref 25060 . . . . . 6  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  ran  R
)
53, 4syl5req 2328 . . . . 5  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  ran  R  =  X )
62, 5sylan2b 461 . . . 4  |-  ( ( Rel  R  /\  (  _I  |`  U. U. R
)  C_  R )  ->  ran  R  =  X )
763adant2 974 . . 3  |-  ( ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R ) 
C_  R )  ->  ran  R  =  X )
81, 7syl6bi 219 . 2  |-  ( R  e. PresetRel  ->  ( R  e. PresetRel  ->  ran  R  =  X ) )
98pm2.43i 43 1  |-  ( R  e. PresetRel  ->  ran  R  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U.cuni 3827   class class class wbr 4023    _I cid 4304   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694  PresetRelcpresetrel 25215
This theorem is referenced by:  pre2befi2  25232  domcnvpre  25233
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-prs 25223
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