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Theorem inisegn0 27140
Description: Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
inisegn0  |-  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) )

Proof of Theorem inisegn0
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  ran  F  ->  A  e.  _V )
2 snprc 3695 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
32biimpi 186 . . . . 5  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
43imaeq2d 5012 . . . 4  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  ( `' F " (/) ) )
5 ima0 5030 . . . 4  |-  ( `' F " (/) )  =  (/)
64, 5syl6eq 2331 . . 3  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  (/) )
76necon1ai 2488 . 2  |-  ( ( `' F " { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2343 . . 3  |-  ( a  =  A  ->  (
a  e.  ran  F  <->  A  e.  ran  F ) )
9 sneq 3651 . . . . 5  |-  ( a  =  A  ->  { a }  =  { A } )
109imaeq2d 5012 . . . 4  |-  ( a  =  A  ->  ( `' F " { a } )  =  ( `' F " { A } ) )
1110neeq1d 2459 . . 3  |-  ( a  =  A  ->  (
( `' F " { a } )  =/=  (/)  <->  ( `' F " { A } )  =/=  (/) ) )
12 abn0 3473 . . . 4  |-  ( { b  |  b F a }  =/=  (/)  <->  E. b 
b F a )
13 vex 2791 . . . . . 6  |-  a  e. 
_V
14 iniseg 5044 . . . . . 6  |-  ( a  e.  _V  ->  ( `' F " { a } )  =  {
b  |  b F a } )
1513, 14ax-mp 8 . . . . 5  |-  ( `' F " { a } )  =  {
b  |  b F a }
1615neeq1i 2456 . . . 4  |-  ( ( `' F " { a } )  =/=  (/)  <->  { b  |  b F a }  =/=  (/) )
1713elrn 4919 . . . 4  |-  ( a  e.  ran  F  <->  E. b 
b F a )
1812, 16, 173bitr4ri 269 . . 3  |-  ( a  e.  ran  F  <->  ( `' F " { a } )  =/=  (/) )
198, 11, 18vtoclbg 2844 . 2  |-  ( A  e.  _V  ->  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) ) )
201, 7, 19pm5.21nii 342 1  |-  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640   class class class wbr 4023   `'ccnv 4688   ran crn 4690   "cima 4692
This theorem is referenced by:  dnnumch3lem  27143  dnnumch3  27144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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