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Theorem 1st2val 6304
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 4869 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5661 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. ) )
3 df-ov 6016 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  x }
v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. )
4 vex 2895 . . . . . . . 8  |-  w  e. 
_V
5 vex 2895 . . . . . . . 8  |-  v  e. 
_V
6 simpl 444 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  x  =  w )
7 mpt2v 6095 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  x )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  x }
87eqcomi 2384 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( x  e.  _V ,  y  e.  _V  |->  x )
96, 8, 4ovmpt2a 6136 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  x } v )  =  w )
104, 5, 9mp2an 654 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  x }
v )  =  w
113, 10eqtr3i 2402 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. )  =  w
122, 11syl6eq 2428 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  w )
134, 5op1std 6289 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 1st `  A
)  =  w )
1412, 13eqtr4d 2415 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
1514exlimivv 1642 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
161, 15sylbi 188 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
17 vex 2895 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2895 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 442 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 a9ev 1663 . . . . . . . . 9  |-  E. z 
z  =  x
2119, 202th 231 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  x )
2221opabbii 4206 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  x }
23 df-xp 4817 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6086 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  =  { <. x ,  y
>.  |  E. z 
z  =  x }
2522, 23, 243eqtr4ri 2411 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  =  ( _V  X.  _V )
2625eleq2i 2444 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5688 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  ->  ( { <. <. x ,  y >. ,  z
>.  |  z  =  x } `  A )  =  (/) )
2826, 27sylnbir 299 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  (/) )
29 dmsnn0 5268 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
3029biimpri 198 . . . . . . 7  |-  ( dom 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2586 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  dom  { A }  =  (/) )
3231unieqd 3961 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  U. (/) )
33 uni0 3977 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2428 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  (/) )
3528, 34eqtr4d 2415 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  U. dom  { A } )
36 1stval 6283 . . 3  |-  ( 1st `  A )  =  U. dom  { A }
3735, 36syl6eqr 2430 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
3816, 37pm2.61i 158 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2543   _Vcvv 2892   (/)c0 3564   {csn 3750   <.cop 3753   U.cuni 3950   {copab 4199    X. cxp 4809   dom cdm 4811   ` cfv 5387  (class class class)co 6013   {coprab 6014    e. cmpt2 6015   1stc1st 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281
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