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Theorem elrn3 28797
Description: Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
Assertion
Ref Expression
elrn3  |-  ( A  e.  ran  B  <->  ( B  i^i  ( _V  X.  { A } ) )  =/=  (/) )

Proof of Theorem elrn3
StepHypRef Expression
1 df-rn 5010 . . 3  |-  ran  B  =  dom  `' B
21eleq2i 2545 . 2  |-  ( A  e.  ran  B  <->  A  e.  dom  `' B )
3 eldm3 28796 . 2  |-  ( A  e.  dom  `' B  <->  ( `' B  |`  { A } )  =/=  (/) )
4 cnvxp 5424 . . . . . . 7  |-  `' ( _V  X.  { A } )  =  ( { A }  X.  _V )
54ineq2i 3697 . . . . . 6  |-  ( `' B  i^i  `' ( _V  X.  { A } ) )  =  ( `' B  i^i  ( { A }  X.  _V ) )
6 cnvin 5413 . . . . . 6  |-  `' ( B  i^i  ( _V 
X.  { A }
) )  =  ( `' B  i^i  `' ( _V  X.  { A } ) )
7 df-res 5011 . . . . . 6  |-  ( `' B  |`  { A } )  =  ( `' B  i^i  ( { A }  X.  _V ) )
85, 6, 73eqtr4ri 2507 . . . . 5  |-  ( `' B  |`  { A } )  =  `' ( B  i^i  ( _V  X.  { A }
) )
98eqeq1i 2474 . . . 4  |-  ( ( `' B  |`  { A } )  =  (/)  <->  `' ( B  i^i  ( _V  X.  { A }
) )  =  (/) )
10 inss2 3719 . . . . . 6  |-  ( B  i^i  ( _V  X.  { A } ) ) 
C_  ( _V  X.  { A } )
11 relxp 5110 . . . . . 6  |-  Rel  ( _V  X.  { A }
)
12 relss 5090 . . . . . 6  |-  ( ( B  i^i  ( _V 
X.  { A }
) )  C_  ( _V  X.  { A }
)  ->  ( Rel  ( _V  X.  { A } )  ->  Rel  ( B  i^i  ( _V  X.  { A }
) ) ) )
1310, 11, 12mp2 9 . . . . 5  |-  Rel  ( B  i^i  ( _V  X.  { A } ) )
14 cnveq0 5463 . . . . 5  |-  ( Rel  ( B  i^i  ( _V  X.  { A }
) )  ->  (
( B  i^i  ( _V  X.  { A }
) )  =  (/)  <->  `' ( B  i^i  ( _V  X.  { A }
) )  =  (/) ) )
1513, 14ax-mp 5 . . . 4  |-  ( ( B  i^i  ( _V 
X.  { A }
) )  =  (/)  <->  `' ( B  i^i  ( _V  X.  { A }
) )  =  (/) )
169, 15bitr4i 252 . . 3  |-  ( ( `' B  |`  { A } )  =  (/)  <->  ( B  i^i  ( _V  X.  { A } ) )  =  (/) )
1716necon3bii 2735 . 2  |-  ( ( `' B  |`  { A } )  =/=  (/)  <->  ( B  i^i  ( _V  X.  { A } ) )  =/=  (/) )
182, 3, 173bitri 271 1  |-  ( A  e.  ran  B  <->  ( B  i^i  ( _V  X.  { A } ) )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   Rel wrel 5004
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011
This theorem is referenced by: (None)
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