Theorem elecALTV 34030

Description: Elementhood in the R-coset of A. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 7785 with this original form of Suppes. Peter Mazsa) (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
elecALTV	|- ( ( A e. V /\ B e. W ) -> ( B e. [ A ] R <-> A R B ) )

Proof of Theorem elecALTV

Step	Hyp	Ref	Expression
1		elimasng 5491	. 2 |- ( ( A e. V /\ B e. W ) -> ( B e. ( R " { A } ) <-> <. A , B >. e. R ) )
2		df-ec 7744	. . 3 |- [ A ] R = ( R " { A } )
3	2	eleq2i 2693	. 2 |- ( B e. [ A ] R <-> B e. ( R " { A } ) )
4		df-br 4654	. 2 |- ( A R B <-> <. A , B >. e. R )
5	1, 3, 4	3bitr4g 303	1 |- ( ( A e. V /\ B e. W ) -> ( B e. [ A ] R <-> A R B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   {csn 4177   <.cop 4183   class class class wbr 4653   "cima 5117   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  eldm4  34037  exan3  34062  exanres3  34064  ecin0  34117