Theorem ercnv 6522

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 6510	. 2 |- ( R Er A -> Rel R )
2		relcnv 4982	. . 3 |- Rel `' R
3		id 19	. . . . . 6 |- ( R Er A -> R Er A )
4	3	ersymb 6515	. . . . 5 |- ( R Er A -> ( y R x <-> x R y ) )
5		vex 2729	. . . . . . 7 |- x e. _V
6		vex 2729	. . . . . . 7 |- y e. _V
7	5, 6	brcnv 4787	. . . . . 6 |- ( x `' R y <-> y R x )
8		df-br 3983	. . . . . 6 |- ( x `' R y <-> <. x , y >. e. `' R )
9	7, 8	bitr3i 185	. . . . 5 |- ( y R x <-> <. x , y >. e. `' R )
10		df-br 3983	. . . . 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 4703	. . 3 |- ( ( ( Rel `' R /\ Rel R ) /\ R Er A ) -> `' R = R )
13	2, 12	mpanl1 431	. 2 |- ( ( Rel R /\ R Er A ) -> `' R = R )
14	1, 13	mpancom 419	1 |- ( R Er A -> `' R = R )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982   `'ccnv 4603   Rel wrel 4609   Er wer 6498

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-er 6501

This theorem is referenced by:  errn  6523