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Theorem fo2nd 5727
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 
_V
2 snexgOLD 3926 . . . . . 6  _V  { }  _V
31, 2ax-mp 7 . . . . 5  { }  _V
43rnex 4542 . . . 4  ran  { }  _V
54uniex 4140 . . 3  U. ran  { }  _V
6 df-2nd 5710 . . 3  2nd  _V  |->  U.
ran  { }
75, 6fnmpti 4970 . 2  2nd  Fn  _V
86rnmpt 4525 . . 3  ran  2nd  {  |  _V  U. ran  { } }
9 vex 2554 . . . . 5 
_V
109, 9opex 3957 . . . . . 6  <. ,  >.  _V
119, 9op2nda 4748 . . . . . . 7  U. ran  {
<. ,  >. }
1211eqcomi 2041 . . . . . 6 
U. ran  { <. ,  >. }
13 sneq 3378 . . . . . . . . . 10  <. , 
>.  { }  { <. , 
>. }
1413rneqd 4506 . . . . . . . . 9  <. , 
>.  ran  { }  ran  { <. ,  >. }
1514unieqd 3582 . . . . . . . 8  <. , 
>.  U. ran  { }  U. ran  { <. , 
>. }
1615eqeq2d 2048 . . . . . . 7  <. , 
>. 
U. ran  { }  U. ran  {
<. ,  >. }
1716rspcev 2650 . . . . . 6 
<. ,  >. 
_V  U.
ran  { <. , 
>. }  _V  U. ran  { }
1810, 12, 17mp2an 402 . . . . 5  _V  U. ran  { }
199, 182th 163 . . . 4  _V  _V  U. ran  { }
2019abbi2i 2149 . . 3  _V  {  |  _V  U. ran  { } }
218, 20eqtr4i 2060 . 2  ran  2nd  _V
22 df-fo 4851 . 2  2nd
: _V -onto-> _V  2nd 
Fn  _V  ran  2nd  _V
237, 21, 22mpbir2an 848 1  2nd : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390   {cab 2023  wrex 2301   _Vcvv 2551   {csn 3367   <.cop 3370   U.cuni 3571   ran crn 4289    Fn wfn 4840   -onto->wfo 4843   2ndc2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-fo 4851  df-2nd 5710
This theorem is referenced by:  2ndcof  5733  2ndexg  5737  df2nd2  5783  2ndconst  5785
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