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Theorem pre2befi2 25335
Description: If  A  <_  B, then  B belongs to the field of the preset. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
Hypothesis
Ref Expression
preoref12.1  |-  X  =  dom  R
Assertion
Ref Expression
pre2befi2  |-  ( ( R  e. PresetRel  /\  A R B )  ->  B  e.  X )

Proof of Theorem pre2befi2
StepHypRef Expression
1 preorel 25328 . . 3  |-  ( R  e. PresetRel  ->  Rel  R )
2 relelrn 4928 . . 3  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
31, 2sylan 457 . 2  |-  ( ( R  e. PresetRel  /\  A R B )  ->  B  e.  ran  R )
4 preoref12.1 . . . 4  |-  X  =  dom  R
54preoran2 25333 . . 3  |-  ( R  e. PresetRel  ->  ran  R  =  X )
65adantr 451 . 2  |-  ( ( R  e. PresetRel  /\  A R B )  ->  ran  R  =  X )
73, 6eleqtrd 2372 1  |-  ( ( R  e. PresetRel  /\  A R B )  ->  B  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705   ran crn 4706   Rel wrel 4710  PresetRelcpresetrel 25318
This theorem is referenced by:  prltub  25363  tolat  25389
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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