Theorem reldvdsr 14225

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.

Step	Hyp	Ref	Expression
1		df-dvdsr 14222	. . 3 |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
2	1	relmptopab 6255	. 2 |- Rel ( ||r ` R )
3		reldvdsr.1	. . 3 |- .|| = ( ||r ` R )
4	3	releqi 4832	. 2 |- ( Rel .|| <-> Rel ( ||r ` R ) )
5	2, 4	mpbir 146	1 |- Rel .||

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2203   E.wrex 2521   _Vcvv 2812   Rel wrel 4753   ` cfv 5351  (class class class)co 6049   Basecbs 13201   .rcmulr 13280   ||_rcdsr 14219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fv 5359  df-dvdsr 14222

This theorem is referenced by:  reldvdsrsrg  14226