Theorem dfrel4v 5584

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6241 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)

Assertion

Ref	Expression
dfrel4v	|- ( Rel R <-> R = { <. x , y >. | x R y } )

Distinct variable group:   x, y, R

Proof of Theorem dfrel4v

Step	Hyp	Ref	Expression
1		dfrel2 5583	. 2 |- ( Rel R <-> `' `' R = R )
2		eqcom 2629	. 2 |- ( `' `' R = R <-> R = `' `' R )
3		cnvcnv3 5582	. . 3 |- `' `' R = { <. x , y >. | x R y }
4	3	eqeq2i 2634	. 2 |- ( R = `' `' R <-> R = { <. x , y >. | x R y } )
5	1, 2, 4	3bitri 286	1 |- ( Rel R <-> R = { <. x , y >. | x R y } )

Syntax hints:   <-> wb 196   = wceq 1483   class class class wbr 4653   {copab 4712   `'ccnv 5113   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-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

This theorem is referenced by:  dfrel4  5585  dffn5  6241  fsplit  7282  pwsle  16152  tgphaus  21920  fneer  32348  inxp2  34129  dfafn5a  41240  sprsymrelfo  41747