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Theorem indpre 25342
Description: The relation induced by a preset on a part of its field is a preset. (Contributed by FL, 28-Dec-2011.)
Assertion
Ref Expression
indpre  |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( R  i^i  ( A  X.  A ) )  e. PresetRel )

Proof of Theorem indpre
StepHypRef Expression
1 sqpre 25341 . 2  |-  ( A  e.  B  ->  ( A  X.  A )  e. PresetRel )
2 int2pre 25340 . 2  |-  ( ( R  e. PresetRel  /\  ( A  X.  A )  e. PresetRel )  ->  ( R  i^i  ( A  X.  A
) )  e. PresetRel )
31, 2sylan2 460 1  |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( R  i^i  ( A  X.  A ) )  e. PresetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    i^i cin 3164    X. cxp 4703  PresetRelcpresetrel 25318
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-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-br 4040  df-opab 4094  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-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-prs 25326
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