MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fo2nd Unicode version

Theorem fo2nd 6140
Description: The  2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo2nd  |-  2nd : _V -onto-> _V

Proof of Theorem fo2nd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4216 . . . . 5  |-  { x }  e.  _V
21rnex 4942 . . . 4  |-  ran  {
x }  e.  _V
32uniex 4516 . . 3  |-  U. ran  { x }  e.  _V
4 df-2nd 6123 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
53, 4fnmpti 5372 . 2  |-  2nd  Fn  _V
64rnmpt 4925 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
7 vex 2791 . . . . 5  |-  y  e. 
_V
8 opex 4237 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op2nda 5157 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
109eqcomi 2287 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
11 sneq 3651 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211rneqd 4906 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1312unieqd 3838 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1413eqeq2d 2294 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1514rspcev 2884 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
168, 10, 15mp2an 653 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
177, 162th 230 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
1817abbi2i 2394 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
196, 18eqtr4i 2306 . 2  |-  ran  2nd  =  _V
20 df-fo 5261 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
215, 19, 20mpbir2an 886 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   {csn 3640   <.cop 3643   U.cuni 3827   ran crn 4690    Fn wfn 5250   -onto->wfo 5253   2ndc2nd 6121
This theorem is referenced by:  2ndcof  6148  df2nd2  6206  2ndconst  6208  iunfo  8161  cdaf  13882  2ndf1  13969  2ndf2  13970  2ndfcl  13972  gsum2d  15223  upxp  17317  uptx  17319  cnmpt2nd  17363  uniiccdif  18933  xppreima  23211  xppreima2  23212  cnre2csqima  23295  br2ndeq  24131  prj3  25080  domrancur1clem  25201  domrancur1c  25202  idval  25725  cmpval  25726  filnetlem4  26330
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-13 1686  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  ax-un 4512
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-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-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-fo 5261  df-2nd 6123
  Copyright terms: Public domain W3C validator