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

Theorem fo2nd 5785
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 2560 . . . . . 6  |-  x  e. 
_V
2 snexgOLD 3935 . . . . . 6  |-  ( x  e.  _V  ->  { x }  e.  _V )
31, 2ax-mp 7 . . . . 5  |-  { x }  e.  _V
43rnex 4599 . . . 4  |-  ran  {
x }  e.  _V
54uniex 4174 . . 3  |-  U. ran  { x }  e.  _V
6 df-2nd 5768 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
75, 6fnmpti 5027 . 2  |-  2nd  Fn  _V
86rnmpt 4582 . . 3  |-  ran  2nd  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
9 vex 2560 . . . . 5  |-  y  e. 
_V
109, 9opex 3966 . . . . . 6  |-  <. y ,  y >.  e.  _V
119, 9op2nda 4805 . . . . . . 7  |-  U. ran  {
<. y ,  y >. }  =  y
1211eqcomi 2044 . . . . . 6  |-  y  = 
U. ran  { <. y ,  y >. }
13 sneq 3386 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1413rneqd 4563 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  ran  { x }  =  ran  { <. y ,  y >. } )
1514unieqd 3591 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  y >. } )
1615eqeq2d 2051 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. ran  { x } 
<->  y  =  U. ran  {
<. y ,  y >. } ) )
1716rspcev 2656 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. ran  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. ran  {
x } )
1810, 12, 17mp2an 402 . . . . 5  |-  E. x  e.  _V  y  =  U. ran  { x }
199, 182th 163 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. ran  { x } )
2019abbi2i 2152 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. ran  { x } }
218, 20eqtr4i 2063 . 2  |-  ran  2nd  =  _V
22 df-fo 4908 . 2  |-  ( 2nd
: _V -onto-> _V  <->  ( 2nd  Fn 
_V  /\  ran  2nd  =  _V ) )
237, 21, 22mpbir2an 849 1  |-  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   {csn 3375   <.cop 3378   U.cuni 3580   ran crn 4346    Fn wfn 4897   -onto->wfo 4900   2ndc2nd 5766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-fo 4908  df-2nd 5768
This theorem is referenced by:  2ndcof  5791  2ndexg  5795  df2nd2  5841  2ndconst  5843
  Copyright terms: Public domain W3C validator