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Theorem inisegn0 29305
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 2979 . 2  |-  ( A  e.  ran  F  ->  A  e.  _V )
2 snprc 3936 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
32biimpi 194 . . . . 5  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
43imaeq2d 5166 . . . 4  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  ( `' F " (/) ) )
5 ima0 5181 . . . 4  |-  ( `' F " (/) )  =  (/)
64, 5syl6eq 2489 . . 3  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  (/) )
76necon1ai 2651 . 2  |-  ( ( `' F " { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2501 . . 3  |-  ( a  =  A  ->  (
a  e.  ran  F  <->  A  e.  ran  F ) )
9 sneq 3884 . . . . 5  |-  ( a  =  A  ->  { a }  =  { A } )
109imaeq2d 5166 . . . 4  |-  ( a  =  A  ->  ( `' F " { a } )  =  ( `' F " { A } ) )
1110neeq1d 2619 . . 3  |-  ( a  =  A  ->  (
( `' F " { a } )  =/=  (/)  <->  ( `' F " { A } )  =/=  (/) ) )
12 abn0 3653 . . . 4  |-  ( { b  |  b F a }  =/=  (/)  <->  E. b 
b F a )
13 vex 2973 . . . . . 6  |-  a  e. 
_V
14 iniseg 5197 . . . . . 6  |-  ( a  e.  _V  ->  ( `' F " { a } )  =  {
b  |  b F a } )
1513, 14ax-mp 5 . . . . 5  |-  ( `' F " { a } )  =  {
b  |  b F a }
1615neeq1i 2616 . . . 4  |-  ( ( `' F " { a } )  =/=  (/)  <->  { b  |  b F a }  =/=  (/) )
1713elrn 5076 . . . 4  |-  ( a  e.  ran  F  <->  E. b 
b F a )
1812, 16, 173bitr4ri 278 . . 3  |-  ( a  e.  ran  F  <->  ( `' F " { a } )  =/=  (/) )
198, 11, 18vtoclbg 3028 . 2  |-  ( A  e.  _V  ->  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) ) )
201, 7, 19pm5.21nii 353 1  |-  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1364   E.wex 1591    e. wcel 1761   {cab 2427    =/= wne 2604   _Vcvv 2970   (/)c0 3634   {csn 3874   class class class wbr 4289   `'ccnv 4835   ran crn 4837   "cima 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849
This theorem is referenced by:  dnnumch3lem  29308  dnnumch3  29309
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