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Theorem df1st2 6425
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 6358 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5647 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  1st  Fn  _V
4 dffn5 5764 . . . . 5  |-  ( 1st 
Fn  _V  <->  1st  =  ( w  e.  _V  |->  ( 1st `  w ) ) )
53, 4mpbi 200 . . . 4  |-  1st  =  ( w  e.  _V  |->  ( 1st `  w ) )
6 mptv 4293 . . . 4  |-  ( w  e.  _V  |->  ( 1st `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
75, 6eqtri 2455 . . 3  |-  1st  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
87reseq1i 5134 . 2  |-  ( 1st  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )
9 resopab 5179 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 1st `  w ) ) }
10 vex 2951 . . . . 5  |-  x  e. 
_V
11 vex 2951 . . . . 5  |-  y  e. 
_V
1210, 11op1std 6349 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 1st `  w
)  =  x )
1312eqeq2d 2446 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 1st `  w
)  <->  z  =  x ) )
1413dfoprab3 6395 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 1st `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  x }
158, 9, 143eqtrri 2460 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   {copab 4257    e. cmpt 4258    X. cxp 4868    |` cres 4872    Fn wfn 5441   -onto->wfo 5444   ` cfv 5446   {coprab 6074   1stc1st 6339
This theorem is referenced by:  df1stres  24083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342
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