Theorem relelec 6553

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 2741	. . . 4 |- ( A e. [ B ] R -> A e. _V )
2		ecexr 6518	. . . 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 4649	. . 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 6551	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
8	4, 6, 7	pm5.21nd 911	1 |- ( Rel R -> ( A e. [ B ] R <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   Rel wrel 4616   [cec 6511

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515

This theorem is referenced by: (None)