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Theorem fsplit 6878
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 6877 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 3109 . . . . 5  |-  x  e. 
_V
2 vex 3109 . . . . 5  |-  y  e. 
_V
31, 2brcnv 5174 . . . 4  |-  ( x `' ( 1st  |`  _I  )
y  <->  y ( 1st  |`  _I  ) x )
41brres 5268 . . . . 5  |-  ( y ( 1st  |`  _I  )
x  <->  ( y 1st x  /\  y  e.  _I  ) )
5 19.42v 1780 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )
)
6 vex 3109 . . . . . . . . . . 11  |-  z  e. 
_V
76, 6op1std 6783 . . . . . . . . . 10  |-  ( y  =  <. z ,  z
>.  ->  ( 1st `  y
)  =  z )
87eqeq1d 2456 . . . . . . . . 9  |-  ( y  =  <. z ,  z
>.  ->  ( ( 1st `  y )  =  x  <-> 
z  =  x ) )
98pm5.32ri 636 . . . . . . . 8  |-  ( ( ( 1st `  y
)  =  x  /\  y  =  <. z ,  z >. )  <->  ( z  =  x  /\  y  =  <. z ,  z
>. ) )
109exbii 1672 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
11 fo1st 6793 . . . . . . . . . 10  |-  1st : _V -onto-> _V
12 fofn 5779 . . . . . . . . . 10  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
1311, 12ax-mp 5 . . . . . . . . 9  |-  1st  Fn  _V
14 fnbrfvb 5888 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  y  e.  _V )  ->  ( ( 1st `  y
)  =  x  <->  y 1st x ) )
1513, 2, 14mp2an 670 . . . . . . . 8  |-  ( ( 1st `  y )  =  x  <->  y 1st x )
16 dfid2 4786 . . . . . . . . . 10  |-  _I  =  { <. z ,  z
>.  |  z  =  z }
1716eleq2i 2532 . . . . . . . . 9  |-  ( y  e.  _I  <->  y  e.  {
<. z ,  z >.  |  z  =  z } )
18 nfe1 1845 . . . . . . . . . . 11  |-  F/ z E. z ( y  =  <. z ,  z
>.  /\  z  =  z )
191819.9 1898 . . . . . . . . . 10  |-  ( E. z E. z ( y  =  <. z ,  z >.  /\  z  =  z )  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
20 elopab 4744 . . . . . . . . . 10  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z E. z ( y  = 
<. z ,  z >.  /\  z  =  z
) )
21 equid 1796 . . . . . . . . . . . 12  |-  z  =  z
2221biantru 503 . . . . . . . . . . 11  |-  ( y  =  <. z ,  z
>. 
<->  ( y  =  <. z ,  z >.  /\  z  =  z ) )
2322exbii 1672 . . . . . . . . . 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 695 . . . . . . 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 4211 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  z >.  =  <. x ,  x >. )
3029eqeq2d 2468 . . . . . . 7  |-  ( z  =  x  ->  (
y  =  <. z ,  z >.  <->  y  =  <. x ,  x >. ) )
311, 30ceqsexv 3143 . . . . . 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 4503 . 2  |-  { <. x ,  y >.  |  x `' ( 1st  |`  _I  )
y }  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
36 relcnv 5362 . . 3  |-  Rel  `' ( 1st  |`  _I  )
37 dfrel4v 5442 . . 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 4531 . 2  |-  ( x  e.  _V  |->  <. x ,  x >. )  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
4035, 38, 393eqtr4i 2493 1  |-  `' ( 1st  |`  _I  )  =  ( x  e. 
_V  |->  <. x ,  x >. )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439   {copab 4496    |-> cmpt 4497    _I cid 4779   `'ccnv 4987    |` cres 4990   Rel wrel 4993    Fn wfn 5565   -onto->wfo 5568   ` cfv 5570   1stc1st 6771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773
This theorem is referenced by: (None)
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