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Theorem fsplit 6920
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 6919 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 3034 . . . . 5  |-  x  e. 
_V
2 vex 3034 . . . . 5  |-  y  e. 
_V
31, 2brcnv 5022 . . . 4  |-  ( x `' ( 1st  |`  _I  )
y  <->  y ( 1st  |`  _I  ) x )
41brres 5117 . . . . 5  |-  ( y ( 1st  |`  _I  )
x  <->  ( y 1st x  /\  y  e.  _I  ) )
5 19.42v 1842 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )
)
6 vex 3034 . . . . . . . . . . 11  |-  z  e. 
_V
76, 6op1std 6822 . . . . . . . . . 10  |-  ( y  =  <. z ,  z
>.  ->  ( 1st `  y
)  =  z )
87eqeq1d 2473 . . . . . . . . 9  |-  ( y  =  <. z ,  z
>.  ->  ( ( 1st `  y )  =  x  <-> 
z  =  x ) )
98pm5.32ri 650 . . . . . . . 8  |-  ( ( ( 1st `  y
)  =  x  /\  y  =  <. z ,  z >. )  <->  ( z  =  x  /\  y  =  <. z ,  z
>. ) )
109exbii 1726 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
11 fo1st 6832 . . . . . . . . . 10  |-  1st : _V -onto-> _V
12 fofn 5808 . . . . . . . . . 10  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
1311, 12ax-mp 5 . . . . . . . . 9  |-  1st  Fn  _V
14 fnbrfvb 5919 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  y  e.  _V )  ->  ( ( 1st `  y
)  =  x  <->  y 1st x ) )
1513, 2, 14mp2an 686 . . . . . . . 8  |-  ( ( 1st `  y )  =  x  <->  y 1st x )
16 dfid2 4757 . . . . . . . . . 10  |-  _I  =  { <. z ,  z
>.  |  z  =  z }
1716eleq2i 2541 . . . . . . . . 9  |-  ( y  e.  _I  <->  y  e.  {
<. z ,  z >.  |  z  =  z } )
18 nfe1 1935 . . . . . . . . . . 11  |-  F/ z E. z ( y  =  <. z ,  z
>.  /\  z  =  z )
191819.9 1990 . . . . . . . . . 10  |-  ( E. z E. z ( y  =  <. z ,  z >.  /\  z  =  z )  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
20 elopab 4709 . . . . . . . . . 10  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z E. z ( y  = 
<. z ,  z >.  /\  z  =  z
) )
21 equid 1863 . . . . . . . . . . . 12  |-  z  =  z
2221biantru 513 . . . . . . . . . . 11  |-  ( y  =  <. z ,  z
>. 
<->  ( y  =  <. z ,  z >.  /\  z  =  z ) )
2322exbii 1726 . . . . . . . . . 10  |-  ( E. z  y  =  <. z ,  z >.  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
2419, 20, 233bitr4i 285 . . . . . . . . 9  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z 
y  =  <. z ,  z >. )
2517, 24bitr2i 258 . . . . . . . 8  |-  ( E. z  y  =  <. z ,  z >.  <->  y  e.  _I  )
2615, 25anbi12i 711 . . . . . . 7  |-  ( ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )  <->  ( y 1st x  /\  y  e.  _I  )
)
275, 10, 263bitr3ri 284 . . . . . 6  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
28 id 22 . . . . . . . . 9  |-  ( z  =  x  ->  z  =  x )
2928, 28opeq12d 4166 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  z >.  =  <. x ,  x >. )
3029eqeq2d 2481 . . . . . . 7  |-  ( z  =  x  ->  (
y  =  <. z ,  z >.  <->  y  =  <. x ,  x >. ) )
311, 30ceqsexv 3070 . . . . . 6  |-  ( E. z ( z  =  x  /\  y  = 
<. z ,  z >.
)  <->  y  =  <. x ,  x >. )
3227, 31bitri 257 . . . . 5  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  y  =  <. x ,  x >. )
334, 32bitri 257 . . . 4  |-  ( y ( 1st  |`  _I  )
x  <->  y  =  <. x ,  x >. )
343, 33bitri 257 . . 3  |-  ( x `' ( 1st  |`  _I  )
y  <->  y  =  <. x ,  x >. )
3534opabbii 4460 . 2  |-  { <. x ,  y >.  |  x `' ( 1st  |`  _I  )
y }  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
36 relcnv 5213 . . 3  |-  Rel  `' ( 1st  |`  _I  )
37 dfrel4v 5293 . . 3  |-  ( Rel  `' ( 1st  |`  _I  )  <->  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y } )
3836, 37mpbi 213 . 2  |-  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y }
39 mptv 4489 . 2  |-  ( x  e.  _V  |->  <. x ,  x >. )  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
4035, 38, 393eqtr4i 2503 1  |-  `' ( 1st  |`  _I  )  =  ( x  e. 
_V  |->  <. x ,  x >. )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   <.cop 3965   class class class wbr 4395   {copab 4453    |-> cmpt 4454    _I cid 4749   `'ccnv 4838    |` cres 4841   Rel wrel 4844    Fn wfn 5584   -onto->wfo 5587   ` cfv 5589   1stc1st 6810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812
This theorem is referenced by: (None)
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