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Theorem df1st2 5868
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 5812 . . . . 5  |-  1st : _V -onto-> _V
2 fofn 5136 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
3 dffn5im 5247 . . . . 5  |-  ( 1st 
Fn  _V  ->  1st  =  ( w  e.  _V  |->  ( 1st `  w ) ) )
41, 2, 3mp2b 8 . . . 4  |-  1st  =  ( w  e.  _V  |->  ( 1st `  w ) )
5 mptv 3881 . . . 4  |-  ( w  e.  _V  |->  ( 1st `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
64, 5eqtri 2076 . . 3  |-  1st  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
76reseq1i 4636 . 2  |-  ( 1st  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )
8 resopab 4680 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 1st `  w ) ) }
9 vex 2577 . . . . 5  |-  x  e. 
_V
10 vex 2577 . . . . 5  |-  y  e. 
_V
119, 10op1std 5803 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 1st `  w
)  =  x )
1211eqeq2d 2067 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 1st `  w
)  <->  z  =  x ) )
1312dfoprab3 5845 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 1st `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  x }
147, 8, 133eqtrri 2081 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3406   {copab 3845    |-> cmpt 3846    X. cxp 4371    |` cres 4375    Fn wfn 4925   -onto->wfo 4928   ` cfv 4930   {coprab 5541   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-sbc 2788  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-res 4385  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-oprab 5544  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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