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

Theorem inisegn0 30584
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 3117 . 2  |-  ( A  e.  ran  F  ->  A  e.  _V )
2 snprc 4086 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
32biimpi 194 . . . . 5  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
43imaeq2d 5330 . . . 4  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  ( `' F " (/) ) )
5 ima0 5345 . . . 4  |-  ( `' F " (/) )  =  (/)
64, 5syl6eq 2519 . . 3  |-  ( -.  A  e.  _V  ->  ( `' F " { A } )  =  (/) )
76necon1ai 2693 . 2  |-  ( ( `' F " { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2534 . . 3  |-  ( a  =  A  ->  (
a  e.  ran  F  <->  A  e.  ran  F ) )
9 sneq 4032 . . . . 5  |-  ( a  =  A  ->  { a }  =  { A } )
109imaeq2d 5330 . . . 4  |-  ( a  =  A  ->  ( `' F " { a } )  =  ( `' F " { A } ) )
1110neeq1d 2739 . . 3  |-  ( a  =  A  ->  (
( `' F " { a } )  =/=  (/)  <->  ( `' F " { A } )  =/=  (/) ) )
12 abn0 3799 . . . 4  |-  ( { b  |  b F a }  =/=  (/)  <->  E. b 
b F a )
13 vex 3111 . . . . . 6  |-  a  e. 
_V
14 iniseg 5361 . . . . . 6  |-  ( a  e.  _V  ->  ( `' F " { a } )  =  {
b  |  b F a } )
1513, 14ax-mp 5 . . . . 5  |-  ( `' F " { a } )  =  {
b  |  b F a }
1615neeq1i 2747 . . . 4  |-  ( ( `' F " { a } )  =/=  (/)  <->  { b  |  b F a }  =/=  (/) )
1713elrn 5236 . . . 4  |-  ( a  e.  ran  F  <->  E. b 
b F a )
1812, 16, 173bitr4ri 278 . . 3  |-  ( a  e.  ran  F  <->  ( `' F " { a } )  =/=  (/) )
198, 11, 18vtoclbg 3167 . 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 1374   E.wex 1591    e. wcel 1762   {cab 2447    =/= wne 2657   _Vcvv 3108   (/)c0 3780   {csn 4022   class class class wbr 4442   `'ccnv 4993   ran crn 4995   "cima 4997
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 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007
This theorem is referenced by:  dnnumch3lem  30587  dnnumch3  30588
  Copyright terms: Public domain W3C validator