Theorem reldvdsr 14104

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 14101	. . 3 |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
2	1	relmptopab 6223	. 2 |- Rel ( ||r ` R )
3		reldvdsr.1	. . 3 |- .|| = ( ||r ` R )
4	3	releqi 4809	. 2 |- ( Rel .|| <-> Rel ( ||r ` R ) )
5	2, 4	mpbir 146	1 |- Rel .||

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   Rel wrel 4730   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   ||_rcdsr 14098

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334  df-dvdsr 14101

This theorem is referenced by:  reldvdsrsrg  14105