MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldvdsr Structured version   Visualization version   Unicode version

Theorem reldvdsr 17950
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 17947 . . 3  |-  ||r  =  (
w  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  w
)  /\  E. z  e.  ( Base `  w
) ( z ( .r `  w ) x )  =  y ) } )
21relmptopab 6536 . 2  |-  Rel  ( ||r `  R )
3 reldvdsr.1 . . 3  |-  .||  =  (
||r `  R )
43releqi 4923 . 2  |-  ( Rel  .|| 
<->  Rel  ( ||r `
 R ) )
52, 4mpbir 214 1  |-  Rel  .||
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   Basecbs 15199   .rcmulr 15269   ||rcdsr 17944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-dvdsr 17947
This theorem is referenced by:  dvdsr  17952  isunit  17963  subrgdvds  18100
  Copyright terms: Public domain W3C validator