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Theorem brwitnlem 7175
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 7168 . . . . 5  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( ( O `  <. A ,  B >. )  e.  _V  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
31, 2mpbiran 918 . . . 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 2688 . . . . 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 2555 . . . 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 4462 . . 3  |-  ( A R B  <->  A ( `' O " ( _V 
\  1o ) ) B )
17 df-br 4457 . . 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 6299 . . 3  |-  ( A O B )  =  ( O `  <. A ,  B >. )
2019neeq1i 2742 . 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468   (/)c0 3793   <.cop 4038   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   1oc1o 7141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6299  df-1o 7148
This theorem is referenced by:  brgic  16444  brric  17520  brlmic  17841  hmph  20403
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