ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fo2nd Unicode version

Theorem fo2nd 6184
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 2755 . . . . . 6  |-  x  e. 
_V
21snex 4203 . . . . 5  |-  { x }  e.  _V
32rnex 4912 . . . 4  |-  ran  {
x }  e.  _V
43uniex 4455 . . 3  |-  U. ran  { x }  e.  _V
5 df-2nd 6167 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
64, 5fnmpti 5363 . 2  |-  2nd  Fn  _V
75rnmpt 4893 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
8 vex 2755 . . . . 5  |-  y  e. 
_V
98, 8opex 4247 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op2nda 5131 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
1110eqcomi 2193 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
12 sneq 3618 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312rneqd 4874 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1413unieqd 3835 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1514eqeq2d 2201 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1615rspcev 2856 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
179, 11, 16mp2an 426 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
188, 172th 174 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
1918abbi2i 2304 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
207, 19eqtr4i 2213 . 2  |-  ran  2nd  =  _V
21 df-fo 5241 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
226, 20, 21mpbir2an 944 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   {cab 2175   E.wrex 2469   _Vcvv 2752   {csn 3607   <.cop 3610   U.cuni 3824   ran crn 4645    Fn wfn 5230   -onto->wfo 5233   2ndc2nd 6165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-fo 5241  df-2nd 6167
This theorem is referenced by:  2ndcof  6190  2ndexg  6194  df2nd2  6246  2ndconst  6248  suplocexprlemmu  7748  suplocexprlemdisj  7750  suplocexprlemloc  7751  suplocexprlemub  7753  upxp  14249  uptx  14251  cnmpt2nd  14266
  Copyright terms: Public domain W3C validator