Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pofunOLD Unicode version

Theorem pofunOLD 26531
Description: A function preserves a partial order relation. (Moved to pofun 4346 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
pofun.1OLD  |-  S  =  { <. x ,  y
>.  |  X R Y }
pofun.2OLD  |-  ( x  =  y  ->  X  =  Y )
Assertion
Ref Expression
pofunOLD  |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
Distinct variable groups:    x, R, y    y, X    x, Y    x, A    x, B
Allowed substitution hints:    A( y)    B( y)    S( x, y)    X( x)    Y( y)

Proof of Theorem pofunOLD
StepHypRef Expression
1 pofun.1OLD . 2  |-  S  =  { <. x ,  y
>.  |  X R Y }
2 pofun.2OLD . 2  |-  ( x  =  y  ->  X  =  Y )
31, 2pofun 4346 1  |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   {copab 4092    Po wpo 4328
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-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-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-po 4330
  Copyright terms: Public domain W3C validator