Theorem brcodir 5170

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 4942	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ z `' R B ) ) )
2		vex 2824	. . . . . 6 |- z e. _V
3		brcnvg 4956	. . . . . 6 |- ( ( z e. _V /\ B e. W ) -> ( z `' R B <-> B R z ) )
4	2, 3	mpan 428	. . . . 5 |- ( B e. W -> ( z `' R B <-> B R z ) )
5	4	anbi2d 468	. . . 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 1878	. 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 1545   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   `'ccnv 4768   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-co 4778

This theorem is referenced by:  codir  5171