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Theorem fo2nd 5929
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 2622 . . . . . 6  |-  x  e. 
_V
21snex 4020 . . . . 5  |-  { x }  e.  _V
32rnex 4700 . . . 4  |-  ran  {
x }  e.  _V
43uniex 4264 . . 3  |-  U. ran  { x }  e.  _V
5 df-2nd 5912 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
64, 5fnmpti 5142 . 2  |-  2nd  Fn  _V
75rnmpt 4683 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
8 vex 2622 . . . . 5  |-  y  e. 
_V
98, 8opex 4056 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op2nda 4915 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
1110eqcomi 2092 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
12 sneq 3457 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312rneqd 4664 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1413unieqd 3664 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1514eqeq2d 2099 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1615rspcev 2722 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
179, 11, 16mp2an 417 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
188, 172th 172 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
1918abbi2i 2202 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
207, 19eqtr4i 2111 . 2  |-  ran  2nd  =  _V
21 df-fo 5021 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
226, 20, 21mpbir2an 888 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619   {csn 3446   <.cop 3449   U.cuni 3653   ran crn 4439    Fn wfn 5010   -onto->wfo 5013   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-fo 5021  df-2nd 5912
This theorem is referenced by:  2ndcof  5935  2ndexg  5939  df2nd2  5985  2ndconst  5987
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