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Theorem sqpre 25238
Description: A square product is a preset. (Contributed by FL, 28-Dec-2011.)
Assertion
Ref Expression
sqpre  |-  ( A  e.  V  ->  ( A  X.  A )  e. PresetRel )

Proof of Theorem sqpre
StepHypRef Expression
1 xpexg 4800 . . 3  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
21anidms 626 . 2  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
3 relxp 4794 . . . 4  |-  Rel  ( A  X.  A )
43a1i 10 . . 3  |-  ( ( A  X.  A )  e.  _V  ->  Rel  ( A  X.  A
) )
5 xpidtr 5065 . . . 4  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
65a1i 10 . . 3  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  o.  ( A  X.  A ) ) 
C_  ( A  X.  A ) )
7 scprefat 25071 . . . 4  |-  (  _I  |`  U. U. ( A  X.  A ) ) 
C_  ( A  X.  A )
87a1i 10 . . 3  |-  ( ( A  X.  A )  e.  _V  ->  (  _I  |`  U. U. ( A  X.  A ) ) 
C_  ( A  X.  A ) )
9 isprsr 25224 . . 3  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  e. PresetRel  <->  ( Rel  ( A  X.  A )  /\  ( ( A  X.  A )  o.  ( A  X.  A ) ) 
C_  ( A  X.  A )  /\  (  _I  |`  U. U. ( A  X.  A ) ) 
C_  ( A  X.  A ) ) ) )
104, 6, 8, 9mpbir3and 1135 . 2  |-  ( ( A  X.  A )  e.  _V  ->  ( A  X.  A )  e. PresetRel )
112, 10syl 15 1  |-  ( A  e.  V  ->  ( A  X.  A )  e. PresetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    C_ wss 3152   U.cuni 3827    _I cid 4304    X. cxp 4687    |` cres 4691    o. ccom 4693   Rel wrel 4694  PresetRelcpresetrel 25215
This theorem is referenced by:  indpre  25239
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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