Theorem relelec 6346

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 2631	. . . 4 |- ( A e. [ B ] R -> A e. _V )
2		ecexr 6311	. . . 4 |- ( A e. [ B ] R -> B e. _V )
3	1, 2	jca 301	. . 3 |- ( A e. [ B ] R -> ( A e. _V /\ B e. _V ) )
4	3	adantl 272	. 2 |- ( ( Rel R /\ A e. [ B ] R ) -> ( A e. _V /\ B e. _V ) )
5		brrelex12 4489	. . 3 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 264	. 2 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7		elecg 6344	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
8	4, 6, 7	pm5.21nd 864	1 |- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1439   _Vcvv 2620   class class class wbr 3851   Rel wrel 4457   [cec 6304

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-ec 6308

This theorem is referenced by: (None)