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Theorem fsplit 6698
Description: A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 6697 in order to build compound functions such as  y  =  ( ( sqr `  x
)  +  ( abs `  x ) ). (Contributed by NM, 17-Sep-2007.)
Assertion
Ref Expression
fsplit  |-  `' ( 1st  |`  _I  )  =  ( x  e. 
_V  |->  <. x ,  x >. )

Proof of Theorem fsplit
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2996 . . . . 5  |-  x  e. 
_V
2 vex 2996 . . . . 5  |-  y  e. 
_V
31, 2brcnv 5043 . . . 4  |-  ( x `' ( 1st  |`  _I  )
y  <->  y ( 1st  |`  _I  ) x )
41brres 5138 . . . . 5  |-  ( y ( 1st  |`  _I  )
x  <->  ( y 1st x  /\  y  e.  _I  ) )
5 19.42v 1924 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )
)
6 vex 2996 . . . . . . . . . . 11  |-  z  e. 
_V
76, 6op1std 6608 . . . . . . . . . 10  |-  ( y  =  <. z ,  z
>.  ->  ( 1st `  y
)  =  z )
87eqeq1d 2451 . . . . . . . . 9  |-  ( y  =  <. z ,  z
>.  ->  ( ( 1st `  y )  =  x  <-> 
z  =  x ) )
98pm5.32ri 638 . . . . . . . 8  |-  ( ( ( 1st `  y
)  =  x  /\  y  =  <. z ,  z >. )  <->  ( z  =  x  /\  y  =  <. z ,  z
>. ) )
109exbii 1634 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
11 fo1st 6617 . . . . . . . . . 10  |-  1st : _V -onto-> _V
12 fofn 5643 . . . . . . . . . 10  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
1311, 12ax-mp 5 . . . . . . . . 9  |-  1st  Fn  _V
14 fnbrfvb 5753 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  y  e.  _V )  ->  ( ( 1st `  y
)  =  x  <->  y 1st x ) )
1513, 2, 14mp2an 672 . . . . . . . 8  |-  ( ( 1st `  y )  =  x  <->  y 1st x )
16 dfid2 4659 . . . . . . . . . 10  |-  _I  =  { <. z ,  z
>.  |  z  =  z }
1716eleq2i 2507 . . . . . . . . 9  |-  ( y  e.  _I  <->  y  e.  {
<. z ,  z >.  |  z  =  z } )
18 nfe1 1778 . . . . . . . . . . 11  |-  F/ z E. z ( y  =  <. z ,  z
>.  /\  z  =  z )
191819.9 1827 . . . . . . . . . 10  |-  ( E. z E. z ( y  =  <. z ,  z >.  /\  z  =  z )  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
20 elopab 4618 . . . . . . . . . 10  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z E. z ( y  = 
<. z ,  z >.  /\  z  =  z
) )
21 equid 1729 . . . . . . . . . . . 12  |-  z  =  z
2221biantru 505 . . . . . . . . . . 11  |-  ( y  =  <. z ,  z
>. 
<->  ( y  =  <. z ,  z >.  /\  z  =  z ) )
2322exbii 1634 . . . . . . . . . 10  |-  ( E. z  y  =  <. z ,  z >.  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
2419, 20, 233bitr4i 277 . . . . . . . . 9  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z 
y  =  <. z ,  z >. )
2517, 24bitr2i 250 . . . . . . . 8  |-  ( E. z  y  =  <. z ,  z >.  <->  y  e.  _I  )
2615, 25anbi12i 697 . . . . . . 7  |-  ( ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )  <->  ( y 1st x  /\  y  e.  _I  )
)
275, 10, 263bitr3ri 276 . . . . . 6  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
28 id 22 . . . . . . . . 9  |-  ( z  =  x  ->  z  =  x )
2928, 28opeq12d 4088 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  z >.  =  <. x ,  x >. )
3029eqeq2d 2454 . . . . . . 7  |-  ( z  =  x  ->  (
y  =  <. z ,  z >.  <->  y  =  <. x ,  x >. ) )
311, 30ceqsexv 3030 . . . . . 6  |-  ( E. z ( z  =  x  /\  y  = 
<. z ,  z >.
)  <->  y  =  <. x ,  x >. )
3227, 31bitri 249 . . . . 5  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  y  =  <. x ,  x >. )
334, 32bitri 249 . . . 4  |-  ( y ( 1st  |`  _I  )
x  <->  y  =  <. x ,  x >. )
343, 33bitri 249 . . 3  |-  ( x `' ( 1st  |`  _I  )
y  <->  y  =  <. x ,  x >. )
3534opabbii 4377 . 2  |-  { <. x ,  y >.  |  x `' ( 1st  |`  _I  )
y }  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
36 relcnv 5227 . . 3  |-  Rel  `' ( 1st  |`  _I  )
37 dfrel4v 5310 . . 3  |-  ( Rel  `' ( 1st  |`  _I  )  <->  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y } )
3836, 37mpbi 208 . 2  |-  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y }
39 mptv 4405 . 2  |-  ( x  e.  _V  |->  <. x ,  x >. )  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
4035, 38, 393eqtr4i 2473 1  |-  `' ( 1st  |`  _I  )  =  ( x  e. 
_V  |->  <. x ,  x >. )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2993   <.cop 3904   class class class wbr 4313   {copab 4370    e. cmpt 4371    _I cid 4652   `'ccnv 4860    |` cres 4863   Rel wrel 4866    Fn wfn 5434   -onto->wfo 5437   ` cfv 5439   1stc1st 6596
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 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fo 5445  df-fv 5447  df-1st 6598
This theorem is referenced by: (None)
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