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Theorem 1st2val 6614
Description: Value of an alternate definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
1st2val  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A )
Distinct variable group:    x, y, z
Allowed substitution hints:    A( x, y, z)

Proof of Theorem 1st2val
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4909 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5703 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. ) )
3 df-ov 6106 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  x }
v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. )
4 vex 2987 . . . . . . . 8  |-  w  e. 
_V
5 vex 2987 . . . . . . . 8  |-  v  e. 
_V
6 simpl 457 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  x  =  w )
7 mpt2v 6192 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  x )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  x }
87eqcomi 2447 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( x  e.  _V ,  y  e.  _V  |->  x )
96, 8, 4ovmpt2a 6233 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  x } v )  =  w )
104, 5, 9mp2an 672 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  x }
v )  =  w
113, 10eqtr3i 2465 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. )  =  w
122, 11syl6eq 2491 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  w )
134, 5op1std 6599 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 1st `  A
)  =  w )
1412, 13eqtr4d 2478 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
1514exlimivv 1689 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
161, 15sylbi 195 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
17 vex 2987 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2987 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 455 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 ax6ev 1710 . . . . . . . . 9  |-  E. z 
z  =  x
2119, 202th 239 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  x )
2221opabbii 4368 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  x }
23 df-xp 4858 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6183 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  =  { <. x ,  y
>.  |  E. z 
z  =  x }
2522, 23, 243eqtr4ri 2474 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  =  ( _V  X.  _V )
2625eleq2i 2507 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5726 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  ->  ( { <. <. x ,  y >. ,  z
>.  |  z  =  x } `  A )  =  (/) )
2826, 27sylnbir 307 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  (/) )
29 dmsnn0 5316 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
3029biimpri 206 . . . . . . 7  |-  ( dom 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2666 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  dom  { A }  =  (/) )
3231unieqd 4113 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  U. (/) )
33 uni0 4130 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2491 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  (/) )
3528, 34eqtr4d 2478 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  U. dom  { A } )
36 1stval 6591 . . 3  |-  ( 1st `  A )  =  U. dom  { A }
3735, 36syl6eqr 2493 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
3816, 37pm2.61i 164 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2618   _Vcvv 2984   (/)c0 3649   {csn 3889   <.cop 3895   U.cuni 4103   {copab 4361    X. cxp 4850   dom cdm 4852   ` cfv 5430  (class class class)co 6103   {coprab 6104    e. cmpt2 6105   1stc1st 6587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589
This theorem is referenced by: (None)
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