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Theorem fo1st 6596
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 4533 . . . . 5  |-  { x }  e.  _V
21dmex 6511 . . . 4  |-  dom  {
x }  e.  _V
32uniex 6376 . . 3  |-  U. dom  { x }  e.  _V
4 df-1st 6577 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
53, 4fnmpti 5539 . 2  |-  1st  Fn  _V
64rnmpt 5085 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
7 vex 2975 . . . . 5  |-  y  e. 
_V
8 opex 4556 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op1sta 5321 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
109eqcomi 2447 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
11 sneq 3887 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211dmeqd 5042 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1312unieqd 4101 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1413eqeq2d 2454 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1514rspcev 3073 . . . . . 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 2554 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
196, 18eqtr4i 2466 . 2  |-  ran  1st  =  _V
20 df-fo 5424 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
215, 19, 20mpbir2an 911 1  |-  1st : _V -onto-> _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2716   _Vcvv 2972   {csn 3877   <.cop 3883   U.cuni 4091   dom cdm 4840   ran crn 4841    Fn wfn 5413   -onto->wfo 5416   1stc1st 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-fun 5420  df-fn 5421  df-fo 5424  df-1st 6577
This theorem is referenced by:  1stcof  6604  df1st2  6659  1stconst  6661  fsplit  6677  algrflem  6681  fpwwe  8813  axpre-sup  9336  homadm  14908  homacd  14909  dmaf  14917  cdaf  14918  1stf1  15002  1stf2  15003  1stfcl  15007  upxp  19196  uptx  19198  cnmpt1st  19241  bcthlem4  20838  uniiccdif  21058  vafval  23981  smfval  23983  0vfval  23984  vsfval  24013  xppreima  25964  xppreima2  25965  1stpreima  26001  cnre2csqima  26341  br1steq  27585
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