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Theorem preoran2 25333
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 25327 . . 3  |-  ( R  e. PresetRel  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
2 idref 5775 . . . . 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 25163 . . . . . 6  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  ran  R
)
53, 4syl5req 2341 . . . . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   U.cuni 3843   class class class wbr 4039    _I cid 4320   dom cdm 4705   ran crn 4706    |` cres 4707    o. ccom 4709   Rel wrel 4710  PresetRelcpresetrel 25318
This theorem is referenced by:  pre2befi2  25335  domcnvpre  25336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-prs 25326
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