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Theorem fo2nd 6360
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 4398 . . . . 5  |-  { x }  e.  _V
21rnex 5126 . . . 4  |-  ran  {
x }  e.  _V
32uniex 4698 . . 3  |-  U. ran  { x }  e.  _V
4 df-2nd 6343 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
53, 4fnmpti 5566 . 2  |-  2nd  Fn  _V
64rnmpt 5109 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
7 vex 2952 . . . . 5  |-  y  e. 
_V
8 opex 4420 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op2nda 5347 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
109eqcomi 2440 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
11 sneq 3818 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211rneqd 5090 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1312unieqd 4019 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1413eqeq2d 2447 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1514rspcev 3045 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
168, 10, 15mp2an 654 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
177, 162th 231 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
1817abbi2i 2547 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
196, 18eqtr4i 2459 . 2  |-  ran  2nd  =  _V
20 df-fo 5453 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
215, 19, 20mpbir2an 887 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2699   _Vcvv 2949   {csn 3807   <.cop 3810   U.cuni 4008   ran crn 4872    Fn wfn 5442   -onto->wfo 5445   2ndc2nd 6341
This theorem is referenced by:  2ndcof  6368  df2nd2  6427  2ndconst  6429  iunfo  8407  cdaf  14198  2ndf1  14285  2ndf2  14286  2ndfcl  14288  gsum2d  15539  upxp  17648  uptx  17650  cnmpt2nd  17694  uniiccdif  19463  xppreima  24052  xppreima2  24053  2ndpreima  24089  cnre2csqima  24302  br2ndeq  25392  filnetlem4  26402
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-fun 5449  df-fn 5450  df-fo 5453  df-2nd 6343
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