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Theorem fo1st 6181
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 4253 . . . . 5  |-  { x }  e.  _V
21dmex 4978 . . . 4  |-  dom  {
x }  e.  _V
32uniex 4553 . . 3  |-  U. dom  { x }  e.  _V
4 df-1st 6164 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
53, 4fnmpti 5409 . 2  |-  1st  Fn  _V
64rnmpt 4962 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
7 vex 2825 . . . . 5  |-  y  e. 
_V
8 opex 4274 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op1sta 5191 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
109eqcomi 2320 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
11 sneq 3685 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211dmeqd 4918 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1312unieqd 3875 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1413eqeq2d 2327 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1514rspcev 2918 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
168, 10, 15mp2an 653 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
177, 162th 230 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
1817abbi2i 2427 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
196, 18eqtr4i 2339 . 2  |-  ran  1st  =  _V
20 df-fo 5298 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
215, 19, 20mpbir2an 886 1  |-  1st : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701   {cab 2302   E.wrex 2578   _Vcvv 2822   {csn 3674   <.cop 3677   U.cuni 3864   dom cdm 4726   ran crn 4727    Fn wfn 5287   -onto->wfo 5290   1stc1st 6162
This theorem is referenced by:  1stcof  6189  df1st2  6247  1stconst  6249  fsplit  6265  algrflem  6266  fpwwe  8313  axpre-sup  8836  homadm  13921  homacd  13922  dmaf  13930  cdaf  13931  1stf1  14015  1stf2  14016  1stfcl  14020  upxp  17373  uptx  17375  cnmpt1st  17418  bcthlem4  18802  uniiccdif  18986  vafval  21214  smfval  21216  0vfval  21217  vsfval  21246  xppreima  23208  xppreima2  23209  1stpreima  23245  cnre2csqima  23378  br1steq  24515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-fun 5294  df-fn 5295  df-fo 5298  df-1st 6164
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