Theorem brcodir 4732

Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Assertion

Ref	Expression
brcodir	|- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ B R z ) ) )

Distinct variable groups:   z, A   z, B   z, R   z, V   z, W

Proof of Theorem brcodir

Step	Hyp	Ref	Expression
1		brcog 4520	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ z `' R B ) ) )
2		vex 2604	. . . . . 6 |- z e. _V
3		brcnvg 4534	. . . . . 6 |- ( ( z e. _V /\ B e. W ) -> ( z `' R B <-> B R z ) )
4	2, 3	mpan 414	. . . . 5 |- ( B e. W -> ( z `' R B <-> B R z ) )
5	4	anbi2d 451	. . . 4 |- ( B e. W -> ( ( A R z /\ z `' R B ) <-> ( A R z /\ B R z ) ) )
6	5	adantl 271	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A R z /\ z `' R B ) <-> ( A R z /\ B R z ) ) )
7	6	exbidv 1746	. 2 |- ( ( A e. V /\ B e. W ) -> ( E. z ( A R z /\ z `' R B ) <-> E. z ( A R z /\ B R z ) ) )
8	1, 7	bitrd 186	1 |- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ B R z ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   `'ccnv 4362   o. ccom 4367

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-v 2603  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-cnv 4371  df-co 4372

This theorem is referenced by:  codir  4733