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Theorem fo1st 6366
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 4405 . . . . 5  |-  { x }  e.  _V
21dmex 5132 . . . 4  |-  dom  {
x }  e.  _V
32uniex 4705 . . 3  |-  U. dom  { x }  e.  _V
4 df-1st 6349 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
53, 4fnmpti 5573 . 2  |-  1st  Fn  _V
64rnmpt 5116 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
7 vex 2959 . . . . 5  |-  y  e. 
_V
8 opex 4427 . . . . . 6  |-  <. y ,  y >.  e.  _V
97, 7op1sta 5351 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
109eqcomi 2440 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
11 sneq 3825 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1211dmeqd 5072 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1312unieqd 4026 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1413eqeq2d 2447 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1514rspcev 3052 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
168, 10, 15mp2an 654 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
177, 162th 231 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
1817abbi2i 2547 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
196, 18eqtr4i 2459 . 2  |-  ran  1st  =  _V
20 df-fo 5460 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
215, 19, 20mpbir2an 887 1  |-  1st : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956   {csn 3814   <.cop 3817   U.cuni 4015   dom cdm 4878   ran crn 4879    Fn wfn 5449   -onto->wfo 5452   1stc1st 6347
This theorem is referenced by:  1stcof  6374  df1st2  6433  1stconst  6435  fsplit  6451  algrflem  6455  fpwwe  8521  axpre-sup  9044  homadm  14195  homacd  14196  dmaf  14204  cdaf  14205  1stf1  14289  1stf2  14290  1stfcl  14294  upxp  17655  uptx  17657  cnmpt1st  17700  bcthlem4  19280  uniiccdif  19470  vafval  22082  smfval  22084  0vfval  22085  vsfval  22114  xppreima  24059  xppreima2  24060  1stpreima  24095  cnre2csqima  24309  br1steq  25398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-fo 5460  df-1st 6349
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