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Theorem inisegn0 35331
Description: Nonemptyness 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 3067 . 2  |-  ( A  e.  ran  F  ->  A  e.  _V )
2 snprc 4034 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
32biimpi 194 . . . . 5  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
43imaeq2d 5156 . . . 4  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  ( `' F " (/) ) )
5 ima0 5171 . . . 4  |-  ( `' F " (/) )  =  (/)
64, 5syl6eq 2459 . . 3  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  (/) )
76necon1ai 2634 . 2  |-  ( ( `' F " { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2474 . . 3  |-  ( a  =  A  ->  (
a  e.  ran  F  <->  A  e.  ran  F ) )
9 sneq 3981 . . . . 5  |-  ( a  =  A  ->  { a }  =  { A } )
109imaeq2d 5156 . . . 4  |-  ( a  =  A  ->  ( `' F " { a } )  =  ( `' F " { A } ) )
1110neeq1d 2680 . . 3  |-  ( a  =  A  ->  (
( `' F " { a } )  =/=  (/)  <->  ( `' F " { A } )  =/=  (/) ) )
12 abn0 3757 . . . 4  |-  ( { b  |  b F a }  =/=  (/)  <->  E. b 
b F a )
13 vex 3061 . . . . . 6  |-  a  e. 
_V
14 iniseg 5187 . . . . . 6  |-  ( a  e.  _V  ->  ( `' F " { a } )  =  {
b  |  b F a } )
1513, 14ax-mp 5 . . . . 5  |-  ( `' F " { a } )  =  {
b  |  b F a }
1615neeq1i 2688 . . . 4  |-  ( ( `' F " { a } )  =/=  (/)  <->  { b  |  b F a }  =/=  (/) )
1713elrn 5063 . . . 4  |-  ( a  e.  ran  F  <->  E. b 
b F a )
1812, 16, 173bitr4ri 278 . . 3  |-  ( a  e.  ran  F  <->  ( `' F " { a } )  =/=  (/) )
198, 11, 18vtoclbg 3117 . 2  |-  ( A  e.  _V  ->  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) ) )
201, 7, 19pm5.21nii 351 1  |-  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387    =/= wne 2598   _Vcvv 3058   (/)c0 3737   {csn 3971   class class class wbr 4394   `'ccnv 4821   ran crn 4823   "cima 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835
This theorem is referenced by:  dnnumch3lem  35334  dnnumch3  35335
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