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Theorem indpre 25239
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 25238 . 2  |-  ( A  e.  B  ->  ( A  X.  A )  e. PresetRel )
2 int2pre 25237 . 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 1684    i^i cin 3151    X. cxp 4687  PresetRelcpresetrel 25215
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-rab 2552  df-v 2790  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-br 4024  df-opab 4078  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-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-prs 25223
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