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Theorem brwitnlem 7169
Description: Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
brwitnlem.r  |-  R  =  ( `' O "
( _V  \  1o ) )
brwitnlem.o  |-  O  Fn  X
Assertion
Ref Expression
brwitnlem  |-  ( A R B  <->  ( A O B )  =/=  (/) )

Proof of Theorem brwitnlem
StepHypRef Expression
1 fvex 5882 . . . . 5  |-  ( O `
 <. A ,  B >. )  e.  _V
2 dif1o 7162 . . . . 5  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( ( O `  <. A ,  B >. )  e.  _V  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
31, 2mpbiran 916 . . . 4  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( O `  <. A ,  B >. )  =/=  (/) )
43anbi2i 694 . . 3  |-  ( (
<. A ,  B >.  e.  X  /\  ( O `
 <. A ,  B >. )  e.  ( _V 
\  1o ) )  <-> 
( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
5 brwitnlem.o . . . 4  |-  O  Fn  X
6 elpreima 6008 . . . 4  |-  ( O  Fn  X  ->  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  e.  ( _V  \  1o ) ) ) )
75, 6ax-mp 5 . . 3  |-  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  e.  ( _V  \  1o ) ) )
8 ndmfv 5896 . . . . . 6  |-  ( -. 
<. A ,  B >.  e. 
dom  O  ->  ( O `
 <. A ,  B >. )  =  (/) )
98necon1ai 2698 . . . . 5  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  dom  O
)
10 fndm 5686 . . . . . 6  |-  ( O  Fn  X  ->  dom  O  =  X )
115, 10ax-mp 5 . . . . 5  |-  dom  O  =  X
129, 11syl6eleq 2565 . . . 4  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  X )
1312pm4.71ri 633 . . 3  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
144, 7, 133bitr4i 277 . 2  |-  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( O `  <. A ,  B >. )  =/=  (/) )
15 brwitnlem.r . . . 4  |-  R  =  ( `' O "
( _V  \  1o ) )
1615breqi 4459 . . 3  |-  ( A R B  <->  A ( `' O " ( _V 
\  1o ) ) B )
17 df-br 4454 . . 3  |-  ( A ( `' O "
( _V  \  1o ) ) B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
1816, 17bitri 249 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
19 df-ov 6298 . . 3  |-  ( A O B )  =  ( O `  <. A ,  B >. )
2019neeq1i 2752 . 2  |-  ( ( A O B )  =/=  (/)  <->  ( O `  <. A ,  B >. )  =/=  (/) )
2114, 18, 203bitr4i 277 1  |-  ( A R B  <->  ( A O B )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    \ cdif 3478   (/)c0 3790   <.cop 4039   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   "cima 5008    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   1oc1o 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6298  df-1o 7142
This theorem is referenced by:  brgic  16189  brric  17264  brlmic  17585  hmph  20145
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