Theorem brcodir 5078

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 4852	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ z `' R B ) ) )
2		vex 2776	. . . . . 6 |- z e. _V
3		brcnvg 4866	. . . . . 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 1849	. 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 1516   e. wcel 2177   _Vcvv 2773   class class class wbr 4050   `'ccnv 4681   o. ccom 4686

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-br 4051  df-opab 4113  df-cnv 4690  df-co 4691

This theorem is referenced by:  codir  5079