Theorem brcodir 5150

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 4922	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ z `' R B ) ) )
2		vex 2816	. . . . . 6 |- z e. _V
3		brcnvg 4936	. . . . . 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 1874	. 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 1541   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   `'ccnv 4748   o. ccom 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-cnv 4757  df-co 4758

This theorem is referenced by:  codir  5151