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Theorem elrn3 29168
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 5000 . . 3  |-  ran  B  =  dom  `' B
21eleq2i 2521 . 2  |-  ( A  e.  ran  B  <->  A  e.  dom  `' B )
3 eldm3 29167 . 2  |-  ( A  e.  dom  `' B  <->  ( `' B  |`  { A } )  =/=  (/) )
4 cnvxp 5414 . . . . . . 7  |-  `' ( _V  X.  { A } )  =  ( { A }  X.  _V )
54ineq2i 3682 . . . . . 6  |-  ( `' B  i^i  `' ( _V  X.  { A } ) )  =  ( `' B  i^i  ( { A }  X.  _V ) )
6 cnvin 5403 . . . . . 6  |-  `' ( B  i^i  ( _V 
X.  { A }
) )  =  ( `' B  i^i  `' ( _V  X.  { A } ) )
7 df-res 5001 . . . . . 6  |-  ( `' B  |`  { A } )  =  ( `' B  i^i  ( { A }  X.  _V ) )
85, 6, 73eqtr4ri 2483 . . . . 5  |-  ( `' B  |`  { A } )  =  `' ( B  i^i  ( _V  X.  { A }
) )
98eqeq1i 2450 . . . 4  |-  ( ( `' B  |`  { A } )  =  (/)  <->  `' ( B  i^i  ( _V  X.  { A }
) )  =  (/) )
10 inss2 3704 . . . . . 6  |-  ( B  i^i  ( _V  X.  { A } ) ) 
C_  ( _V  X.  { A } )
11 relxp 5100 . . . . . 6  |-  Rel  ( _V  X.  { A }
)
12 relss 5080 . . . . . 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 5453 . . . . 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 2711 . 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 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990    |` cres 4991   Rel wrel 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001
This theorem is referenced by: (None)
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