Theorem dfrel4 5585

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.) (Revised by Thierry Arnoux, 11-May-2017.)

Hypotheses

Ref	Expression
dfrel4.1	|- F/_ x R
dfrel4.2	|- F/_ y R

Assertion

Ref	Expression
dfrel4	|- ( Rel R <-> R = { <. x , y >. | x R y } )

Distinct variable group:   x, y

Allowed substitution hints:   R( x, y)

Proof of Theorem dfrel4

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrel4v 5584	. 2 |- ( Rel R <-> R = { <. a , b >. | a R b } )
2		nfcv 2764	. . . . 5 |- F/_ x a
3		dfrel4.1	. . . . 5 |- F/_ x R
4		nfcv 2764	. . . . 5 |- F/_ x b
5	2, 3, 4	nfbr 4699	. . . 4 |- F/ x a R b
6		nfcv 2764	. . . . 5 |- F/_ y a
7		dfrel4.2	. . . . 5 |- F/_ y R
8		nfcv 2764	. . . . 5 |- F/_ y b
9	6, 7, 8	nfbr 4699	. . . 4 |- F/ y a R b
10		nfv 1843	. . . 4 |- F/ a x R y
11		nfv 1843	. . . 4 |- F/ b x R y
12		breq12 4658	. . . 4 |- ( ( a = x /\ b = y ) -> ( a R b <-> x R y ) )
13	5, 9, 10, 11, 12	cbvopab 4721	. . 3 |- { <. a , b >. | a R b } = { <. x , y >. | x R y }
14	13	eqeq2i 2634	. 2 |- ( R = { <. a , b >. | a R b } <-> R = { <. x , y >. | x R y } )
15	1, 14	bitri 264	1 |- ( Rel R <-> R = { <. x , y >. | x R y } )

Syntax hints:   <-> wb 196   = wceq 1483   F/_wnfc 2751   class class class wbr 4653   {copab 4712   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:  feqmptdf  6251