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Theorem fo1st 6815
Description: The  1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo1st  |-  1st : _V -onto-> _V

Proof of Theorem fo1st
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4694 . . . . 5  |-  { x }  e.  _V
21dmex 6728 . . . 4  |-  dom  {
x }  e.  _V
32uniex 6591 . . 3  |-  U. dom  { x }  e.  _V
4 df-1st 6795 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
53, 4fnmpti 5715 . 2  |-  1st  Fn  _V
64rnmpt 5254 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
7 vex 3121 . . . . 5  |-  y  e. 
_V
8 opex 4717 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op1sta 5496 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
109eqcomi 2480 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
11 sneq 4043 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211dmeqd 5211 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1312unieqd 4261 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1413eqeq2d 2481 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1514rspcev 3219 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
168, 10, 15mp2an 672 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
177, 162th 239 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
1817abbi2i 2600 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
196, 18eqtr4i 2499 . 2  |-  ran  1st  =  _V
20 df-fo 5600 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
215, 19, 20mpbir2an 918 1  |-  1st : _V -onto-> _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   _Vcvv 3118   {csn 4033   <.cop 4039   U.cuni 4251   dom cdm 5005   ran crn 5006    Fn wfn 5589   -onto->wfo 5592   1stc1st 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-fun 5596  df-fn 5597  df-fo 5600  df-1st 6795
This theorem is referenced by:  1stcof  6823  df1st2  6881  1stconst  6883  fsplit  6900  algrflem  6904  fpwwe  9036  axpre-sup  9558  homadm  15242  homacd  15243  dmaf  15251  cdaf  15252  1stf1  15336  1stf2  15337  1stfcl  15341  upxp  19992  uptx  19994  cnmpt1st  20037  bcthlem4  21634  uniiccdif  21855  vafval  25319  smfval  25321  0vfval  25322  vsfval  25351  xppreima  27310  xppreima2  27311  1stpreima  27347  cnre2csqima  27718  br1steq  29131
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