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Theorem brwitnlem 7220
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 5891 . . . . 5  |-  ( O `
 <. A ,  B >. )  e.  _V
2 dif1o 7213 . . . . 5  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( ( O `  <. A ,  B >. )  e.  _V  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
31, 2mpbiran 926 . . . 4  |-  ( ( O `  <. A ,  B >. )  e.  ( _V  \  1o )  <-> 
( O `  <. A ,  B >. )  =/=  (/) )
43anbi2i 698 . . 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 6017 . . . 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 5905 . . . . . 6  |-  ( -. 
<. A ,  B >.  e. 
dom  O  ->  ( O `
 <. A ,  B >. )  =  (/) )
98necon1ai 2651 . . . . 5  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  dom  O
)
10 fndm 5693 . . . . . 6  |-  ( O  Fn  X  ->  dom  O  =  X )
115, 10ax-mp 5 . . . . 5  |-  dom  O  =  X
129, 11syl6eleq 2517 . . . 4  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  ->  <. A ,  B >.  e.  X )
1312pm4.71ri 637 . . 3  |-  ( ( O `  <. A ,  B >. )  =/=  (/)  <->  ( <. A ,  B >.  e.  X  /\  ( O `  <. A ,  B >. )  =/=  (/) ) )
144, 7, 133bitr4i 280 . 2  |-  ( <. A ,  B >.  e.  ( `' O "
( _V  \  1o ) )  <->  ( O `  <. A ,  B >. )  =/=  (/) )
15 brwitnlem.r . . . 4  |-  R  =  ( `' O "
( _V  \  1o ) )
1615breqi 4429 . . 3  |-  ( A R B  <->  A ( `' O " ( _V 
\  1o ) ) B )
17 df-br 4424 . . 3  |-  ( A ( `' O "
( _V  \  1o ) ) B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
1816, 17bitri 252 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  ( `' O " ( _V 
\  1o ) ) )
19 df-ov 6308 . . 3  |-  ( A O B )  =  ( O `  <. A ,  B >. )
2019neeq1i 2705 . 2  |-  ( ( A O B )  =/=  (/)  <->  ( O `  <. A ,  B >. )  =/=  (/) )
2114, 18, 203bitr4i 280 1  |-  ( A R B  <->  ( A O B )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080    \ cdif 3433   (/)c0 3761   <.cop 4004   class class class wbr 4423   `'ccnv 4852   dom cdm 4853   "cima 4856    Fn wfn 5596   ` cfv 5601  (class class class)co 6305   1oc1o 7186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ov 6308  df-1o 7193
This theorem is referenced by:  brgic  16932  brric  17971  brlmic  18290  hmph  20789
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