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Theorem fo2nd 6299
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 4339 . . . . 5  |-  { x }  e.  _V
21rnex 5066 . . . 4  |-  ran  {
x }  e.  _V
32uniex 4638 . . 3  |-  U. ran  { x }  e.  _V
4 df-2nd 6282 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
53, 4fnmpti 5506 . 2  |-  2nd  Fn  _V
64rnmpt 5049 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
7 vex 2895 . . . . 5  |-  y  e. 
_V
8 opex 4361 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op2nda 5287 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
109eqcomi 2384 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
11 sneq 3761 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211rneqd 5030 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1312unieqd 3961 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1413eqeq2d 2391 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1514rspcev 2988 . . . . . 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 2491 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
196, 18eqtr4i 2403 . 2  |-  ran  2nd  =  _V
20 df-fo 5393 . 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 1649    e. wcel 1717   {cab 2366   E.wrex 2643   _Vcvv 2892   {csn 3750   <.cop 3753   U.cuni 3950   ran crn 4812    Fn wfn 5382   -onto->wfo 5385   2ndc2nd 6280
This theorem is referenced by:  2ndcof  6307  df2nd2  6366  2ndconst  6368  iunfo  8340  cdaf  14125  2ndf1  14212  2ndf2  14213  2ndfcl  14215  gsum2d  15466  upxp  17569  uptx  17571  cnmpt2nd  17615  uniiccdif  19330  xppreima  23894  xppreima2  23895  2ndpreima  23930  cnre2csqima  24106  br2ndeq  25148  filnetlem4  26094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-fun 5389  df-fn 5390  df-fo 5393  df-2nd 6282
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