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Theorem reldvdsr 15426
Description: The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypothesis
Ref Expression
reldvdsr.1  |-  .||  =  (
||r `  R )
Assertion
Ref Expression
reldvdsr  |-  Rel  .||

Proof of Theorem reldvdsr
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvdsr 15423 . . 3  |-  ||r  =  (
w  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  w
)  /\  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y ) } )
21relmptopab 6065 . 2  |-  Rel  ( ||r `  R )
3 reldvdsr.1 . . 3  |-  .||  =  (
||r `  R )
43releqi 4772 . 2  |-  ( Rel  .|| 
<->  Rel  ( ||r `
 R ) )
52, 4mpbir 200 1  |-  Rel  .||
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   ||rcdsr 15420
This theorem is referenced by:  dvdsr  15428  isunit  15439  subrgdvds  15559
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
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-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fv 5263  df-dvdsr 15423
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