Theorem reldvdsr 14258

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 14255	. . 3 |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } )
2	1	relmptopab 6258	. 2 |- Rel ( ||r ` R )
3		reldvdsr.1	. . 3 |- .|| = ( ||r ` R )
4	3	releqi 4835	. 2 |- ( Rel .|| <-> Rel ( ||r ` R ) )
5	2, 4	mpbir 146	1 |- Rel .||

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   _Vcvv 2815   Rel wrel 4756   ` cfv 5354  (class class class)co 6052   Basecbs 13233   .rcmulr 13312   ||_rcdsr 14252

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 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fv 5362  df-dvdsr 14255

This theorem is referenced by:  reldvdsrsrg  14259