Theorem dfrel6 34115

Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)

Assertion

Ref	Expression
dfrel6	|- ( Rel R <-> ( R i^i ( dom R X. ran R ) ) = R )

Proof of Theorem dfrel6

Step	Hyp	Ref	Expression
1		dfrel5 34114	. 2 |- ( Rel R <-> ( R |` dom R ) = R )
2		dfres3 5403	. . 3 |- ( R |` dom R ) = ( R i^i ( dom R X. ran R ) )
3	2	eqeq1i 2627	. 2 |- ( ( R |` dom R ) = R <-> ( R i^i ( dom R X. ran R ) ) = R )
4	1, 3	bitri 264	1 |- ( Rel R <-> ( R i^i ( dom R X. ran R ) ) = R )

Syntax hints:   <-> wb 196   = wceq 1483   i^i cin 3573   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by: (None)