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Theorem 2nd2val 6365
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 4928 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5720 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. ) )
3 df-ov 6076 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )
4 vex 2951 . . . . . . . 8  |-  w  e. 
_V
5 vex 2951 . . . . . . . 8  |-  v  e. 
_V
6 simpr 448 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  y  =  v )
7 mpt2v 6155 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  y }
87eqcomi 2439 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( x  e.  _V ,  y  e.  _V  |->  y )
96, 8, 5ovmpt2a 6196 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  y } v )  =  v )
104, 5, 9mp2an 654 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  v
113, 10eqtr3i 2457 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )  =  v
122, 11syl6eq 2483 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  v )
134, 5op2ndd 6350 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 2nd `  A
)  =  v )
1412, 13eqtr4d 2470 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
1514exlimivv 1645 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
161, 15sylbi 188 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
17 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 442 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 a9ev 1668 . . . . . . . . 9  |-  E. z 
z  =  y
2119, 202th 231 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  y )
2221opabbii 4264 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  y }
23 df-xp 4876 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6146 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  { <. x ,  y >.  |  E. z  z  =  y }
2522, 23, 243eqtr4ri 2466 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  ( _V  X.  _V )
2625eleq2i 2499 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5747 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  (/) )
2826, 27sylnbir 299 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  (/) )
29 rnsnn0 5328 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
3029biimpri 198 . . . . . . 7  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2641 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
3231unieqd 4018 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
33 uni0 4034 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2483 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
3528, 34eqtr4d 2470 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  = 
U. ran  { A } )
36 2ndval 6344 . . 3  |-  ( 2nd `  A )  =  U. ran  { A }
3735, 36syl6eqr 2485 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
3816, 37pm2.61i 158 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   {csn 3806   <.cop 3809   U.cuni 4007   {copab 4257    X. cxp 4868   dom cdm 4870   ran crn 4871   ` cfv 5446  (class class class)co 6073   {coprab 6074    e. cmpt2 6075   2ndc2nd 6340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342
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