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Theorem fsplit 6442
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 6441 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 2951 . . . . 5  |-  x  e. 
_V
2 vex 2951 . . . . 5  |-  y  e. 
_V
31, 2brcnv 5046 . . . 4  |-  ( x `' ( 1st  |`  _I  )
y  <->  y ( 1st  |`  _I  ) x )
41brres 5143 . . . . 5  |-  ( y ( 1st  |`  _I  )
x  <->  ( y 1st x  /\  y  e.  _I  ) )
5 19.42v 1928 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )
)
6 vex 2951 . . . . . . . . . . 11  |-  z  e. 
_V
76, 6op1std 6348 . . . . . . . . . 10  |-  ( y  =  <. z ,  z
>.  ->  ( 1st `  y
)  =  z )
87eqeq1d 2443 . . . . . . . . 9  |-  ( y  =  <. z ,  z
>.  ->  ( ( 1st `  y )  =  x  <-> 
z  =  x ) )
98pm5.32ri 620 . . . . . . . 8  |-  ( ( ( 1st `  y
)  =  x  /\  y  =  <. z ,  z >. )  <->  ( z  =  x  /\  y  =  <. z ,  z
>. ) )
109exbii 1592 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
11 fo1st 6357 . . . . . . . . . 10  |-  1st : _V -onto-> _V
12 fofn 5646 . . . . . . . . . 10  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
1311, 12ax-mp 8 . . . . . . . . 9  |-  1st  Fn  _V
14 fnbrfvb 5758 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  y  e.  _V )  ->  ( ( 1st `  y
)  =  x  <->  y 1st x ) )
1513, 2, 14mp2an 654 . . . . . . . 8  |-  ( ( 1st `  y )  =  x  <->  y 1st x )
16 dfid2 4492 . . . . . . . . . 10  |-  _I  =  { <. z ,  z
>.  |  z  =  z }
1716eleq2i 2499 . . . . . . . . 9  |-  ( y  e.  _I  <->  y  e.  {
<. z ,  z >.  |  z  =  z } )
18 nfe1 1747 . . . . . . . . . . 11  |-  F/ z E. z ( y  =  <. z ,  z
>.  /\  z  =  z )
191819.9 1797 . . . . . . . . . 10  |-  ( E. z E. z ( y  =  <. z ,  z >.  /\  z  =  z )  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
20 elopab 4454 . . . . . . . . . 10  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z E. z ( y  = 
<. z ,  z >.  /\  z  =  z
) )
21 equid 1688 . . . . . . . . . . . 12  |-  z  =  z
2221biantru 492 . . . . . . . . . . 11  |-  ( y  =  <. z ,  z
>. 
<->  ( y  =  <. z ,  z >.  /\  z  =  z ) )
2322exbii 1592 . . . . . . . . . 10  |-  ( E. z  y  =  <. z ,  z >.  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
2419, 20, 233bitr4i 269 . . . . . . . . 9  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z 
y  =  <. z ,  z >. )
2517, 24bitr2i 242 . . . . . . . 8  |-  ( E. z  y  =  <. z ,  z >.  <->  y  e.  _I  )
2615, 25anbi12i 679 . . . . . . 7  |-  ( ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )  <->  ( y 1st x  /\  y  e.  _I  )
)
275, 10, 263bitr3ri 268 . . . . . 6  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
28 id 20 . . . . . . . . 9  |-  ( z  =  x  ->  z  =  x )
2928, 28opeq12d 3984 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  z >.  =  <. x ,  x >. )
3029eqeq2d 2446 . . . . . . 7  |-  ( z  =  x  ->  (
y  =  <. z ,  z >.  <->  y  =  <. x ,  x >. ) )
311, 30ceqsexv 2983 . . . . . 6  |-  ( E. z ( z  =  x  /\  y  = 
<. z ,  z >.
)  <->  y  =  <. x ,  x >. )
3227, 31bitri 241 . . . . 5  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  y  =  <. x ,  x >. )
334, 32bitri 241 . . . 4  |-  ( y ( 1st  |`  _I  )
x  <->  y  =  <. x ,  x >. )
343, 33bitri 241 . . 3  |-  ( x `' ( 1st  |`  _I  )
y  <->  y  =  <. x ,  x >. )
3534opabbii 4264 . 2  |-  { <. x ,  y >.  |  x `' ( 1st  |`  _I  )
y }  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
36 relcnv 5233 . . 3  |-  Rel  `' ( 1st  |`  _I  )
37 dfrel4v 5313 . . 3  |-  ( Rel  `' ( 1st  |`  _I  )  <->  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y } )
3836, 37mpbi 200 . 2  |-  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y }
39 mptv 4293 . 2  |-  ( x  e.  _V  |->  <. x ,  x >. )  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
4035, 38, 393eqtr4i 2465 1  |-  `' ( 1st  |`  _I  )  =  ( x  e. 
_V  |->  <. x ,  x >. )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   {copab 4257    e. cmpt 4258    _I cid 4485   `'ccnv 4868    |` cres 4871   Rel wrel 4874    Fn wfn 5440   -onto->wfo 5443   ` cfv 5445   1stc1st 6338
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fo 5451  df-fv 5453  df-1st 6340
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