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Theorem fo2nd 6820
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 4697 . . . . 5  |-  { x }  e.  _V
21rnex 6733 . . . 4  |-  ran  {
x }  e.  _V
32uniex 6595 . . 3  |-  U. ran  { x }  e.  _V
4 df-2nd 6800 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
53, 4fnmpti 5715 . 2  |-  2nd  Fn  _V
64rnmpt 5258 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
7 vex 3112 . . . . 5  |-  y  e. 
_V
8 opex 4720 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op2nda 5499 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
109eqcomi 2470 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
11 sneq 4042 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211rneqd 5240 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1312unieqd 4261 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1413eqeq2d 2471 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1514rspcev 3210 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
168, 10, 15mp2an 672 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
177, 162th 239 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
1817abbi2i 2590 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
196, 18eqtr4i 2489 . 2  |-  ran  2nd  =  _V
20 df-fo 5600 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
215, 19, 20mpbir2an 920 1  |-  2nd : _V -onto-> _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   _Vcvv 3109   {csn 4032   <.cop 4038   U.cuni 4251   ran crn 5009    Fn wfn 5589   -onto->wfo 5592   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-fo 5600  df-2nd 6800
This theorem is referenced by:  2ndcof  6828  df2nd2  6886  2ndconst  6888  iunfo  8931  cdaf  15456  2ndf1  15591  2ndf2  15592  2ndfcl  15594  gsum2dlem2  17125  gsum2dOLD  17127  upxp  20250  uptx  20252  cnmpt2nd  20296  uniiccdif  22113  xppreima  27635  xppreima2  27636  2ndpreima  27681  gsummpt2d  27932  cnre2csqima  28054  br2ndeq  29422  filnetlem4  30404
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