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Theorem df1st2 6109
Description: An alternate possible definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
df1st2  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
Distinct variable group:    x, y, z

Proof of Theorem df1st2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fo1st 6048 . . . . 5  |-  1st : _V -onto-> _V
2 fofn 5342 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
3 dffn5im 5460 . . . . 5  |-  ( 1st 
Fn  _V  ->  1st  =  ( w  e.  _V  |->  ( 1st `  w ) ) )
41, 2, 3mp2b 8 . . . 4  |-  1st  =  ( w  e.  _V  |->  ( 1st `  w ) )
5 mptv 4020 . . . 4  |-  ( w  e.  _V  |->  ( 1st `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
64, 5eqtri 2158 . . 3  |-  1st  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
76reseq1i 4810 . 2  |-  ( 1st  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )
8 resopab 4858 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 1st `  w ) ) }
9 vex 2684 . . . . 5  |-  x  e. 
_V
10 vex 2684 . . . . 5  |-  y  e. 
_V
119, 10op1std 6039 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 1st `  w
)  =  x )
1211eqeq2d 2149 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 1st `  w
)  <->  z  =  x ) )
1312dfoprab3 6082 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 1st `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  x }
147, 8, 133eqtrri 2163 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2681   <.cop 3525   {copab 3983    |-> cmpt 3984    X. cxp 4532    |` cres 4536    Fn wfn 5113   -onto->wfo 5116   ` cfv 5118   {coprab 5768   1stc1st 6029
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-oprab 5771  df-1st 6031  df-2nd 6032
This theorem is referenced by: (None)
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