Theorem ercnv 6766

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 6754	. 2 |- ( R Er A -> Rel R )
2		relcnv 5121	. . 3 |- Rel `' R
3		id 19	. . . . . 6 |- ( R Er A -> R Er A )
4	3	ersymb 6759	. . . . 5 |- ( R Er A -> ( y R x <-> x R y ) )
5		vex 2806	. . . . . . 7 |- x e. _V
6		vex 2806	. . . . . . 7 |- y e. _V
7	5, 6	brcnv 4919	. . . . . 6 |- ( x `' R y <-> y R x )
8		df-br 4094	. . . . . 6 |- ( x `' R y <-> <. x , y >. e. `' R )
9	7, 8	bitr3i 186	. . . . 5 |- ( y R x <-> <. x , y >. e. `' R )
10		df-br 4094	. . . . 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 4831	. . 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 1398   e. wcel 2202   <.cop 3676   class class class wbr 4093   `'ccnv 4730   Rel wrel 4736   Er wer 6742

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-er 6745

This theorem is referenced by:  errn  6767