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Theorem brwitnlem 6687
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 5682 . . . . 5  |-  ( O `
 <. A ,  B >. )  e.  _V
2 dif1o 6680 . . . . 5  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( ( O `  <. A ,  B >. )  e.  _V  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
31, 2mpbiran 885 . . . 4  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( O `  <. A ,  B >. )  =/=  (/) )
43anbi2i 676 . . 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 5789 . . . 4  |-  ( O  Fn  X  ->  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  e.  ( _V  \  1o ) ) ) )
75, 6ax-mp 8 . . 3  |-  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  e.  ( _V  \  1o ) ) )
8 ndmfv 5695 . . . . . 6  |-  ( -. 
<. A ,  B >.  e. 
dom  O  ->  ( O `
 <. A ,  B >. )  =  (/) )
98necon1ai 2592 . . . . 5  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  dom  O
)
10 fndm 5484 . . . . . 6  |-  ( O  Fn  X  ->  dom  O  =  X )
115, 10ax-mp 8 . . . . 5  |-  dom  O  =  X
129, 11syl6eleq 2477 . . . 4  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  X )
1312pm4.71ri 615 . . 3  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
144, 7, 133bitr4i 269 . 2  |-  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( O `  <. A ,  B >. )  =/=  (/) )
15 brwitnlem.r . . . 4  |-  R  =  ( `' O "
( _V  \  1o ) )
1615breqi 4159 . . 3  |-  ( A R B  <->  A ( `' O " ( _V 
\  1o ) ) B )
17 df-br 4154 . . 3  |-  ( A ( `' O "
( _V  \  1o ) ) B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
1816, 17bitri 241 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
19 df-ov 6023 . . 3  |-  ( A O B )  =  ( O `  <. A ,  B >. )
2019neeq1i 2560 . 2  |-  ( ( A O B )  =/=  (/)  <->  ( O `  <. A ,  B >. )  =/=  (/) )
2114, 18, 203bitr4i 269 1  |-  ( A R B  <->  ( A O B )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    \ cdif 3260   (/)c0 3571   <.cop 3760   class class class wbr 4153   `'ccnv 4817   dom cdm 4818   "cima 4821    Fn wfn 5389   ` cfv 5394  (class class class)co 6020   1oc1o 6653
This theorem is referenced by:  brgic  14983  brlmic  16067  hmph  17729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-ov 6023  df-1o 6660
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