Theorem ercnv 6832

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 6820	. 2 |- ( R Er A -> Rel R )
2		relcnv 5168	. . 3 |- Rel `' R
3		id 19	. . . . . 6 |- ( R Er A -> R Er A )
4	3	ersymb 6825	. . . . 5 |- ( R Er A -> ( y R x <-> x R y ) )
5		vex 2824	. . . . . . 7 |- x e. _V
6		vex 2824	. . . . . . 7 |- y e. _V
7	5, 6	brcnv 4965	. . . . . 6 |- ( x `' R y <-> y R x )
8		df-br 4132	. . . . . 6 |- ( x `' R y <-> <. x , y >. e. `' R )
9	7, 8	bitr3i 186	. . . . 5 |- ( y R x <-> <. x , y >. e. `' R )
10		df-br 4132	. . . . 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 4876	. . 3 |- ( ( ( Rel `' R /\ Rel R ) /\ R Er A ) -> `' R = R )
13	2, 12	mpanl1 438	. 2 |- ( ( Rel R /\ R Er A ) -> `' R = R )
14	1, 13	mpancom 426	1 |- ( R Er A -> `' R = R )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4131   `'ccnv 4775   Rel wrel 4781   Er wer 6808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-er 6811

This theorem is used by:  errn  6833