Theorem relelec 6541

Description: Membership in an equivalence class when R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)

Assertion

Ref	Expression
relelec	|- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Proof of Theorem relelec

Step	Hyp	Ref	Expression
1		elex 2737	. . . 4 |- ( A e. [ B ] R -> A e. _V )
2		ecexr 6506	. . . 4 |- ( A e. [ B ] R -> B e. _V )
3	1, 2	jca 304	. . 3 |- ( A e. [ B ] R -> ( A e. _V /\ B e. _V ) )
4	3	adantl 275	. 2 |- ( ( Rel R /\ A e. [ B ] R ) -> ( A e. _V /\ B e. _V ) )
5		brrelex12 4642	. . 3 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 265	. 2 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7		elecg 6539	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
8	4, 6, 7	pm5.21nd 906	1 |- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   Rel wrel 4609   [cec 6499

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-sbc 2952  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-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503

This theorem is referenced by: (None)