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Theorem fo1st 5812
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 2577 . . . . . 6  |-  x  e. 
_V
2 snexgOLD 3963 . . . . . 6  |-  ( x  e.  _V  ->  { x }  e.  _V )
31, 2ax-mp 7 . . . . 5  |-  { x }  e.  _V
43dmex 4626 . . . 4  |-  dom  {
x }  e.  _V
54uniex 4202 . . 3  |-  U. dom  { x }  e.  _V
6 df-1st 5795 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
75, 6fnmpti 5055 . 2  |-  1st  Fn  _V
86rnmpt 4610 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
9 vex 2577 . . . . 5  |-  y  e. 
_V
109, 9opex 3994 . . . . . 6  |-  <. y ,  y >.  e.  _V
119, 9op1sta 4830 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
1211eqcomi 2060 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
13 sneq 3414 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1413dmeqd 4565 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1514unieqd 3619 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1615eqeq2d 2067 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1716rspcev 2673 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
1810, 12, 17mp2an 410 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
199, 182th 167 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
2019abbi2i 2168 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
218, 20eqtr4i 2079 . 2  |-  ran  1st  =  _V
22 df-fo 4936 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
237, 21, 22mpbir2an 860 1  |-  1st : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574   {csn 3403   <.cop 3406   U.cuni 3608   dom cdm 4373   ran crn 4374    Fn wfn 4925   -onto->wfo 4928   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-fo 4936  df-1st 5795
This theorem is referenced by:  1stcof  5818  1stexg  5822  df1st2  5868  1stconst  5870  algrflem  5878  algrflemg  5879
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