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Theorem fo1st 6023
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 vex 2663 . . . . . 6  |-  x  e. 
_V
21snex 4079 . . . . 5  |-  { x }  e.  _V
32dmex 4775 . . . 4  |-  dom  {
x }  e.  _V
43uniex 4329 . . 3  |-  U. dom  { x }  e.  _V
5 df-1st 6006 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
64, 5fnmpti 5221 . 2  |-  1st  Fn  _V
75rnmpt 4757 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
8 vex 2663 . . . . 5  |-  y  e. 
_V
98, 8opex 4121 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op1sta 4990 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
1110eqcomi 2121 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
12 sneq 3508 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312dmeqd 4711 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1413unieqd 3717 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1514eqeq2d 2129 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1615rspcev 2763 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
179, 11, 16mp2an 422 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
188, 172th 173 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
1918abbi2i 2232 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
207, 19eqtr4i 2141 . 2  |-  ran  1st  =  _V
21 df-fo 5099 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
226, 20, 21mpbir2an 911 1  |-  1st : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660   {csn 3497   <.cop 3500   U.cuni 3706   dom cdm 4509   ran crn 4510    Fn wfn 5088   -onto->wfo 5091   1stc1st 6004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-fo 5099  df-1st 6006
This theorem is referenced by:  1stcof  6029  1stexg  6033  df1st2  6084  1stconst  6086  algrflem  6094  algrflemg  6095  suplocexprlemell  7489  suplocexprlem2b  7490  suplocexprlemlub  7500  upxp  12368  uptx  12370  cnmpt1st  12384
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