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Theorem fo2nd 6022
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 vex 2661 . . . . . 6  |-  x  e. 
_V
21snex 4077 . . . . 5  |-  { x }  e.  _V
32rnex 4774 . . . 4  |-  ran  {
x }  e.  _V
43uniex 4327 . . 3  |-  U. ran  { x }  e.  _V
5 df-2nd 6005 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
64, 5fnmpti 5219 . 2  |-  2nd  Fn  _V
75rnmpt 4755 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
8 vex 2661 . . . . 5  |-  y  e. 
_V
98, 8opex 4119 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op2nda 4991 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
1110eqcomi 2119 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
12 sneq 3506 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312rneqd 4736 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1413unieqd 3715 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1514eqeq2d 2127 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1615rspcev 2761 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
179, 11, 16mp2an 420 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
188, 172th 173 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
1918abbi2i 2230 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
207, 19eqtr4i 2139 . 2  |-  ran  2nd  =  _V
21 df-fo 5097 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
226, 20, 21mpbir2an 909 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   {cab 2101   E.wrex 2392   _Vcvv 2658   {csn 3495   <.cop 3498   U.cuni 3704   ran crn 4508    Fn wfn 5086   -onto->wfo 5089   2ndc2nd 6003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-fo 5097  df-2nd 6005
This theorem is referenced by:  2ndcof  6028  2ndexg  6032  df2nd2  6083  2ndconst  6085  suplocexprlemmu  7490  suplocexprlemdisj  7492  suplocexprlemloc  7493  suplocexprlemub  7495  upxp  12336  uptx  12338  cnmpt2nd  12353
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