Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrn3 Structured version   Unicode version

Theorem elrn3 27578
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 4856 . . 3  |-  ran  B  =  dom  `' B
21eleq2i 2507 . 2  |-  ( A  e.  ran  B  <->  A  e.  dom  `' B )
3 eldm3 27577 . 2  |-  ( A  e.  dom  `' B  <->  ( `' B  |`  { A } )  =/=  (/) )
4 cnvxp 5260 . . . . . . 7  |-  `' ( _V  X.  { A } )  =  ( { A }  X.  _V )
54ineq2i 3554 . . . . . 6  |-  ( `' B  i^i  `' ( _V  X.  { A } ) )  =  ( `' B  i^i  ( { A }  X.  _V ) )
6 cnvin 5249 . . . . . 6  |-  `' ( B  i^i  ( _V 
X.  { A }
) )  =  ( `' B  i^i  `' ( _V  X.  { A } ) )
7 df-res 4857 . . . . . 6  |-  ( `' B  |`  { A } )  =  ( `' B  i^i  ( { A }  X.  _V ) )
85, 6, 73eqtr4ri 2474 . . . . 5  |-  ( `' B  |`  { A } )  =  `' ( B  i^i  ( _V  X.  { A }
) )
98eqeq1i 2450 . . . 4  |-  ( ( `' B  |`  { A } )  =  (/)  <->  `' ( B  i^i  ( _V  X.  { A }
) )  =  (/) )
10 inss2 3576 . . . . . 6  |-  ( B  i^i  ( _V  X.  { A } ) ) 
C_  ( _V  X.  { A } )
11 relxp 4952 . . . . . 6  |-  Rel  ( _V  X.  { A }
)
12 relss 4932 . . . . . 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 5299 . . . . 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 2645 . 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 1369    e. wcel 1756    =/= wne 2611   _Vcvv 2977    i^i cin 3332    C_ wss 3333   (/)c0 3642   {csn 3882    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   Rel wrel 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator