Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inisegn0 Unicode version

Theorem inisegn0 27055
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 2956 . 2  |-  ( A  e.  ran  F  ->  A  e.  _V )
2 snprc 3863 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
32biimpi 187 . . . . 5  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
43imaeq2d 5194 . . . 4  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  ( `' F " (/) ) )
5 ima0 5212 . . . 4  |-  ( `' F " (/) )  =  (/)
64, 5syl6eq 2483 . . 3  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  (/) )
76necon1ai 2640 . 2  |-  ( ( `' F " { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2495 . . 3  |-  ( a  =  A  ->  (
a  e.  ran  F  <->  A  e.  ran  F ) )
9 sneq 3817 . . . . 5  |-  ( a  =  A  ->  { a }  =  { A } )
109imaeq2d 5194 . . . 4  |-  ( a  =  A  ->  ( `' F " { a } )  =  ( `' F " { A } ) )
1110neeq1d 2611 . . 3  |-  ( a  =  A  ->  (
( `' F " { a } )  =/=  (/)  <->  ( `' F " { A } )  =/=  (/) ) )
12 abn0 3638 . . . 4  |-  ( { b  |  b F a }  =/=  (/)  <->  E. b 
b F a )
13 vex 2951 . . . . . 6  |-  a  e. 
_V
14 iniseg 5226 . . . . . 6  |-  ( a  e.  _V  ->  ( `' F " { a } )  =  {
b  |  b F a } )
1513, 14ax-mp 8 . . . . 5  |-  ( `' F " { a } )  =  {
b  |  b F a }
1615neeq1i 2608 . . . 4  |-  ( ( `' F " { a } )  =/=  (/)  <->  { b  |  b F a }  =/=  (/) )
1713elrn 5101 . . . 4  |-  ( a  e.  ran  F  <->  E. b 
b F a )
1812, 16, 173bitr4ri 270 . . 3  |-  ( a  e.  ran  F  <->  ( `' F " { a } )  =/=  (/) )
198, 11, 18vtoclbg 3004 . 2  |-  ( A  e.  _V  ->  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) ) )
201, 7, 19pm5.21nii 343 1  |-  ( A  e.  ran  F  <->  ( `' F " { A }
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   _Vcvv 2948   (/)c0 3620   {csn 3806   class class class wbr 4204   `'ccnv 4868   ran crn 4870   "cima 4872
This theorem is referenced by:  dnnumch3lem  27058  dnnumch3  27059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882
  Copyright terms: Public domain W3C validator