Theorem relelec 6169

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 2610	. . . 4 |- ( A e. [ B ] R -> A e. _V )
2		ecexr 6134	. . . 4 |- ( A e. [ B ] R -> B e. _V )
3	1, 2	jca 300	. . 3 |- ( A e. [ B ] R -> ( A e. _V /\ B e. _V ) )
4	3	adantl 271	. 2 |- ( ( Rel R /\ A e. [ B ] R ) -> ( A e. _V /\ B e. _V ) )
5		brrelex12 4399	. . 3 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 263	. 2 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7		elecg 6167	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
8	4, 6, 7	pm5.21nd 858	1 |- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   Rel wrel 4368   [cec 6127

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131

This theorem is referenced by: (None)