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Theorem brwitnlem 7196
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 5858 . . . . 5  |-  ( O `
 <. A ,  B >. )  e.  _V
2 dif1o 7189 . . . . 5  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( ( O `  <. A ,  B >. )  e.  _V  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
31, 2mpbiran 929 . . . 4  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( O `  <. A ,  B >. )  =/=  (/) )
43anbi2i 705 . . 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 5986 . . . 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 5872 . . . . . 6  |-  ( -. 
<. A ,  B >.  e. 
dom  O  ->  ( O `
 <. A ,  B >. )  =  (/) )
98necon1ai 2651 . . . . 5  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  dom  O
)
10 fndm 5657 . . . . . 6  |-  ( O  Fn  X  ->  dom  O  =  X )
115, 10ax-mp 5 . . . . 5  |-  dom  O  =  X
129, 11syl6eleq 2540 . . . 4  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  X )
1312pm4.71ri 643 . . 3  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
144, 7, 133bitr4i 285 . 2  |-  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( O `  <. A ,  B >. )  =/=  (/) )
15 brwitnlem.r . . . 4  |-  R  =  ( `' O "
( _V  \  1o ) )
1615breqi 4380 . . 3  |-  ( A R B  <->  A ( `' O " ( _V 
\  1o ) ) B )
17 df-br 4375 . . 3  |-  ( A ( `' O "
( _V  \  1o ) ) B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
1816, 17bitri 257 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
19 df-ov 6279 . . 3  |-  ( A O B )  =  ( O `  <. A ,  B >. )
2019neeq1i 2688 . 2  |-  ( ( A O B )  =/=  (/)  <->  ( O `  <. A ,  B >. )  =/=  (/) )
2114, 18, 203bitr4i 285 1  |-  ( A R B  <->  ( A O B )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1448    e. wcel 1891    =/= wne 2622   _Vcvv 3013    \ cdif 3369   (/)c0 3699   <.cop 3942   class class class wbr 4374   `'ccnv 4811   dom cdm 4812   "cima 4815    Fn wfn 5556   ` cfv 5561  (class class class)co 6276   1oc1o 7162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-fv 5569  df-ov 6279  df-1o 7169
This theorem is referenced by:  brgic  16944  brric  17983  brlmic  18302  hmph  20802
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