Theorem brcodir 5057

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 4833	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ z `' R B ) ) )
2		vex 2766	. . . . . 6 |- z e. _V
3		brcnvg 4847	. . . . . 6 |- ( ( z e. _V /\ B e. W ) -> ( z `' R B <-> B R z ) )
4	2, 3	mpan 424	. . . . 5 |- ( B e. W -> ( z `' R B <-> B R z ) )
5	4	anbi2d 464	. . . 4 |- ( B e. W -> ( ( A R z /\ z `' R B ) <-> ( A R z /\ B R z ) ) )
6	5	adantl 277	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A R z /\ z `' R B ) <-> ( A R z /\ B R z ) ) )
7	6	exbidv 1839	. 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 188	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 104   <-> wb 105   E.wex 1506   e. wcel 2167   _Vcvv 2763   class class class wbr 4033   `'ccnv 4662   o. ccom 4667

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-co 4672

This theorem is referenced by:  codir  5058