Theorem relelec 6720

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 2811	. . . 4 |- ( A e. [ B ] R -> A e. _V )
2		ecexr 6683	. . . 4 |- ( A e. [ B ] R -> B e. _V )
3	1, 2	jca 306	. . 3 |- ( A e. [ B ] R -> ( A e. _V /\ B e. _V ) )
4	3	adantl 277	. 2 |- ( ( Rel R /\ A e. [ B ] R ) -> ( A e. _V /\ B e. _V ) )
5		brrelex12 4756	. . 3 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 267	. 2 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7		elecg 6718	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
8	4, 6, 7	pm5.21nd 921	1 |- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   _Vcvv 2799   class class class wbr 4082   Rel wrel 4723   [cec 6676

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-sbc 3029  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-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-ec 6680

This theorem is referenced by:  eqgid  13758  eqg0el  13761