MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2nd2val Structured version   Unicode version

Theorem 2nd2val 6602
Description: Value of an alternate definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
2nd2val  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Distinct variable group:    x, y, z
Allowed substitution hints:    A( x, y, z)

Proof of Theorem 2nd2val
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4896 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5690 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. ) )
3 df-ov 6093 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )
4 vex 2974 . . . . . . . 8  |-  w  e. 
_V
5 vex 2974 . . . . . . . 8  |-  v  e. 
_V
6 simpr 461 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  y  =  v )
7 mpt2v 6179 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  y }
87eqcomi 2446 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( x  e.  _V ,  y  e.  _V  |->  y )
96, 8, 5ovmpt2a 6220 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  y } v )  =  v )
104, 5, 9mp2an 672 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  v
113, 10eqtr3i 2464 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )  =  v
122, 11syl6eq 2490 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  v )
134, 5op2ndd 6587 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 2nd `  A
)  =  v )
1412, 13eqtr4d 2477 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
1514exlimivv 1689 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
161, 15sylbi 195 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
17 vex 2974 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2974 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 455 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 ax6ev 1710 . . . . . . . . 9  |-  E. z 
z  =  y
2119, 202th 239 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  y )
2221opabbii 4355 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  y }
23 df-xp 4845 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6170 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  { <. x ,  y >.  |  E. z  z  =  y }
2522, 23, 243eqtr4ri 2473 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  ( _V  X.  _V )
2625eleq2i 2506 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5713 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  (/) )
2826, 27sylnbir 307 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  (/) )
29 rnsnn0 5304 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
3029biimpri 206 . . . . . . 7  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2653 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
3231unieqd 4100 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
33 uni0 4117 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2490 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
3528, 34eqtr4d 2477 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  = 
U. ran  { A } )
36 2ndval 6579 . . 3  |-  ( 2nd `  A )  =  U. ran  { A }
3735, 36syl6eqr 2492 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
3816, 37pm2.61i 164 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2605   _Vcvv 2971   (/)c0 3636   {csn 3876   <.cop 3882   U.cuni 4090   {copab 4348    X. cxp 4837   dom cdm 4839   ran crn 4840   ` cfv 5417  (class class class)co 6090   {coprab 6091    e. cmpt2 6092   2ndc2nd 6575
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-2nd 6577
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator