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Theorem fo2nd 5816
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 2605 . . . . . 6  |-  x  e. 
_V
21snex 3965 . . . . 5  |-  { x }  e.  _V
32rnex 4627 . . . 4  |-  ran  {
x }  e.  _V
43uniex 4200 . . 3  |-  U. ran  { x }  e.  _V
5 df-2nd 5799 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
64, 5fnmpti 5058 . 2  |-  2nd  Fn  _V
75rnmpt 4610 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
8 vex 2605 . . . . 5  |-  y  e. 
_V
98, 8opex 3992 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op2nda 4835 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
1110eqcomi 2086 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
12 sneq 3417 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312rneqd 4591 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1413unieqd 3620 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1514eqeq2d 2093 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1615rspcev 2702 . . . . . 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 2194 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
207, 19eqtr4i 2105 . 2  |-  ran  2nd  =  _V
21 df-fo 4938 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
226, 20, 21mpbir2an 884 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   {cab 2068   E.wrex 2350   _Vcvv 2602   {csn 3406   <.cop 3409   U.cuni 3609   ran crn 4372    Fn wfn 4927   -onto->wfo 4930   2ndc2nd 5797
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-fo 4938  df-2nd 5799
This theorem is referenced by:  2ndcof  5822  2ndexg  5826  df2nd2  5872  2ndconst  5874
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