Theorem ercnv 6699

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 6687	. 2 |- ( R Er A -> Rel R )
2		relcnv 5105	. . 3 |- Rel `' R
3		id 19	. . . . . 6 |- ( R Er A -> R Er A )
4	3	ersymb 6692	. . . . 5 |- ( R Er A -> ( y R x <-> x R y ) )
5		vex 2802	. . . . . . 7 |- x e. _V
6		vex 2802	. . . . . . 7 |- y e. _V
7	5, 6	brcnv 4904	. . . . . 6 |- ( x `' R y <-> y R x )
8		df-br 4083	. . . . . 6 |- ( x `' R y <-> <. x , y >. e. `' R )
9	7, 8	bitr3i 186	. . . . 5 |- ( y R x <-> <. x , y >. e. `' R )
10		df-br 4083	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
11	4, 9, 10	3bitr3g 222	. . . 4 |- ( R Er A -> ( <. x , y >. e. `' R <-> <. x , y >. e. R ) )
12	11	eqrelrdv2 4817	. . 3 |- ( ( ( Rel `' R /\ Rel R ) /\ R Er A ) -> `' R = R )
13	2, 12	mpanl1 434	. 2 |- ( ( Rel R /\ R Er A ) -> `' R = R )
14	1, 13	mpancom 422	1 |- ( R Er A -> `' R = R )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4082   `'ccnv 4717   Rel wrel 4723   Er wer 6675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-er 6678

This theorem is referenced by:  errn  6700