Theorem ercnv 6443

Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
ercnv	|- ( R Er A -> `' R = R )

Proof of Theorem ercnv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		errel 6431	. 2 |- ( R Er A -> Rel R )
2		relcnv 4912	. . 3 |- Rel `' R
3		id 19	. . . . . 6 |- ( R Er A -> R Er A )
4	3	ersymb 6436	. . . . 5 |- ( R Er A -> ( y R x <-> x R y ) )
5		vex 2684	. . . . . . 7 |- x e. _V
6		vex 2684	. . . . . . 7 |- y e. _V
7	5, 6	brcnv 4717	. . . . . 6 |- ( x `' R y <-> y R x )
8		df-br 3925	. . . . . 6 |- ( x `' R y <-> <. x , y >. e. `' R )
9	7, 8	bitr3i 185	. . . . 5 |- ( y R x <-> <. x , y >. e. `' R )
10		df-br 3925	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
11	4, 9, 10	3bitr3g 221	. . . 4 |- ( R Er A -> ( <. x , y >. e. `' R <-> <. x , y >. e. R ) )
12	11	eqrelrdv2 4633	. . 3 |- ( ( ( Rel `' R /\ Rel R ) /\ R Er A ) -> `' R = R )
13	2, 12	mpanl1 430	. 2 |- ( ( Rel R /\ R Er A ) -> `' R = R )
14	1, 13	mpancom 418	1 |- ( R Er A -> `' R = R )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   <.cop 3525   class class class wbr 3924   `'ccnv 4533   Rel wrel 4539   Er wer 6419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-er 6422

This theorem is referenced by:  errn  6444