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Theorem df1st2 6869
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 6804 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5797 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 dffn5 5913 . . . . 5  |-  ( 1st 
Fn  _V  <->  1st  =  ( w  e.  _V  |->  ( 1st `  w ) ) )
53, 4mpbi 208 . . . 4  |-  1st  =  ( w  e.  _V  |->  ( 1st `  w ) )
6 mptv 4539 . . . 4  |-  ( w  e.  _V  |->  ( 1st `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
75, 6eqtri 2496 . . 3  |-  1st  =  { <. w ,  z
>.  |  z  =  ( 1st `  w ) }
87reseq1i 5269 . 2  |-  ( 1st  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )
9 resopab 5320 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 1st `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 1st `  w ) ) }
10 vex 3116 . . . . 5  |-  x  e. 
_V
11 vex 3116 . . . . 5  |-  y  e. 
_V
1210, 11op1std 6794 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 1st `  w
)  =  x )
1312eqeq2d 2481 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 1st `  w
)  <->  z  =  x ) )
1413dfoprab3 6840 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 1st `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  x }
158, 9, 143eqtrri 2501 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   {copab 4504    |-> cmpt 4505    X. cxp 4997    |` cres 5001    Fn wfn 5583   -onto->wfo 5586   ` cfv 5588   {coprab 6285   1stc1st 6782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-oprab 6288  df-1st 6784  df-2nd 6785
This theorem is referenced by:  df1stres  27222
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