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Theorem fo1st 5910
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 2622 . . . . . 6  |-  x  e. 
_V
21snex 4011 . . . . 5  |-  { x }  e.  _V
32dmex 4687 . . . 4  |-  dom  {
x }  e.  _V
43uniex 4255 . . 3  |-  U. dom  { x }  e.  _V
5 df-1st 5893 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
64, 5fnmpti 5128 . 2  |-  1st  Fn  _V
75rnmpt 4671 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
8 vex 2622 . . . . 5  |-  y  e. 
_V
98, 8opex 4047 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op1sta 4899 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
1110eqcomi 2092 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
12 sneq 3452 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312dmeqd 4626 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1413unieqd 3659 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1514eqeq2d 2099 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1615rspcev 2722 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
179, 11, 16mp2an 417 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
188, 172th 172 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
1918abbi2i 2202 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
207, 19eqtr4i 2111 . 2  |-  ran  1st  =  _V
21 df-fo 5008 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
226, 20, 21mpbir2an 888 1  |-  1st : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619   {csn 3441   <.cop 3444   U.cuni 3648   dom cdm 4428   ran crn 4429    Fn wfn 4997   -onto->wfo 5000   1stc1st 5891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-fo 5008  df-1st 5893
This theorem is referenced by:  1stcof  5916  1stexg  5920  df1st2  5966  1stconst  5968  algrflem  5976  algrflemg  5977
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