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Theorem 1st2val 6408
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 4971 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5763 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. ) )
3 df-ov 6120 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  x }
v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  <. w ,  v
>. )
4 vex 2968 . . . . . . . 8  |-  w  e. 
_V
5 vex 2968 . . . . . . . 8  |-  v  e. 
_V
6 simpl 445 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  x  =  w )
7 mpt2v 6199 . . . . . . . . . 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 6240 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  x } v )  =  w )
104, 5, 9mp2an 655 . . . . . . 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 6393 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 1st `  A
)  =  w )
1412, 13eqtr4d 2478 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
1514exlimivv 1647 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
161, 15sylbi 189 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A ) )
17 vex 2968 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2968 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 443 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 a9ev 1671 . . . . . . . . 9  |-  E. z 
z  =  x
2119, 202th 232 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  x )
2221opabbii 4303 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  x }
23 df-xp 4919 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6190 . . . . . . 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 5786 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  x }  ->  ( { <. <. x ,  y >. ,  z
>.  |  z  =  x } `  A )  =  (/) )
2826, 27sylnbir 300 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  (/) )
29 dmsnn0 5370 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
3029biimpri 199 . . . . . . 7  |-  ( dom 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2654 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  dom  { A }  =  (/) )
3231unieqd 4055 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  U. (/) )
33 uni0 4071 . . . . 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 6387 . . 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 159 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360   E.wex 1551    = wceq 1654    e. wcel 1728    =/= wne 2606   _Vcvv 2965   (/)c0 3616   {csn 3843   <.cop 3846   U.cuni 4044   {copab 4296    X. cxp 4911   dom cdm 4913   ` cfv 5489  (class class class)co 6117   {coprab 6118    e. cmpt2 6119   1stc1st 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5453  df-fun 5491  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385
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