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Theorem df1st2 6084
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 6023 . . . . 5  |-  1st : _V -onto-> _V
2 fofn 5317 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
3 dffn5im 5435 . . . . 5  |-  ( 1st 
Fn  _V  ->  1st  =  ( w  e.  _V  |->  ( 1st `  w ) ) )
41, 2, 3mp2b 8 . . . 4  |-  1st  =  ( w  e.  _V  |->  ( 1st `  w ) )
5 mptv 3995 . . . 4  |-  ( w  e.  _V  |->  ( 1st `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
64, 5eqtri 2138 . . 3  |-  1st  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
76reseq1i 4785 . 2  |-  ( 1st  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )
8 resopab 4833 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 1st `  w ) ) }
9 vex 2663 . . . . 5  |-  x  e. 
_V
10 vex 2663 . . . . 5  |-  y  e. 
_V
119, 10op1std 6014 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 1st `  w
)  =  x )
1211eqeq2d 2129 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 1st `  w
)  <->  z  =  x ) )
1312dfoprab3 6057 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 1st `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  x }
147, 8, 133eqtrri 2143 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   _Vcvv 2660   <.cop 3500   {copab 3958    |-> cmpt 3959    X. cxp 4507    |` cres 4511    Fn wfn 5088   -onto->wfo 5091   ` cfv 5093   {coprab 5743   1stc1st 6004
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-oprab 5746  df-1st 6006  df-2nd 6007
This theorem is referenced by: (None)
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