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Theorem fo1st 6152
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 2740 . . . . . 6  |-  x  e. 
_V
21snex 4182 . . . . 5  |-  { x }  e.  _V
32dmex 4889 . . . 4  |-  dom  {
x }  e.  _V
43uniex 4434 . . 3  |-  U. dom  { x }  e.  _V
5 df-1st 6135 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
64, 5fnmpti 5340 . 2  |-  1st  Fn  _V
75rnmpt 4871 . . 3  |-  ran  1st  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
8 vex 2740 . . . . 5  |-  y  e. 
_V
98, 8opex 4226 . . . . . 6  |-  <. y ,  y >.  e.  _V
108, 8op1sta 5106 . . . . . . 7  |-  U. dom  {
<. y ,  y >. }  =  y
1110eqcomi 2181 . . . . . 6  |-  y  = 
U. dom  { <. y ,  y >. }
12 sneq 3602 . . . . . . . . . 10  |-  ( x  =  <. y ,  y
>.  ->  { x }  =  { <. y ,  y
>. } )
1312dmeqd 4825 . . . . . . . . 9  |-  ( x  =  <. y ,  y
>.  ->  dom  { x }  =  dom  { <. y ,  y >. } )
1413unieqd 3818 . . . . . . . 8  |-  ( x  =  <. y ,  y
>.  ->  U. dom  { x }  =  U. dom  { <. y ,  y >. } )
1514eqeq2d 2189 . . . . . . 7  |-  ( x  =  <. y ,  y
>.  ->  ( y  = 
U. dom  { x } 
<->  y  =  U. dom  {
<. y ,  y >. } ) )
1615rspcev 2841 . . . . . 6  |-  ( (
<. y ,  y >.  e.  _V  /\  y  = 
U. dom  { <. y ,  y >. } )  ->  E. x  e.  _V  y  =  U. dom  {
x } )
179, 11, 16mp2an 426 . . . . 5  |-  E. x  e.  _V  y  =  U. dom  { x }
188, 172th 174 . . . 4  |-  ( y  e.  _V  <->  E. x  e.  _V  y  =  U. dom  { x } )
1918abbi2i 2292 . . 3  |-  _V  =  { y  |  E. x  e.  _V  y  =  U. dom  { x } }
207, 19eqtr4i 2201 . 2  |-  ran  1st  =  _V
21 df-fo 5218 . 2  |-  ( 1st
: _V -onto-> _V  <->  ( 1st  Fn 
_V  /\  ran  1st  =  _V ) )
226, 20, 21mpbir2an 942 1  |-  1st : _V -onto-> _V
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2737   {csn 3591   <.cop 3594   U.cuni 3807   dom cdm 4623   ran crn 4624    Fn wfn 5207   -onto->wfo 5210   1stc1st 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-fun 5214  df-fn 5215  df-fo 5218  df-1st 6135
This theorem is referenced by:  1stcof  6158  1stexg  6162  df1st2  6214  1stconst  6216  algrflem  6224  algrflemg  6225  suplocexprlemell  7703  suplocexprlem2b  7704  suplocexprlemlub  7714  upxp  13439  uptx  13441  cnmpt1st  13455
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