MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsplit Unicode version

Theorem fsplit 6351
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 6350 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 2876 . . . . 5  |-  x  e. 
_V
2 vex 2876 . . . . 5  |-  y  e. 
_V
31, 2brcnv 4967 . . . 4  |-  ( x `' ( 1st  |`  _I  )
y  <->  y ( 1st  |`  _I  ) x )
41brres 5064 . . . . 5  |-  ( y ( 1st  |`  _I  )
x  <->  ( y 1st x  /\  y  e.  _I  ) )
5 19.42v 1915 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )
)
6 vex 2876 . . . . . . . . . . 11  |-  z  e. 
_V
76, 6op1std 6257 . . . . . . . . . 10  |-  ( y  =  <. z ,  z
>.  ->  ( 1st `  y
)  =  z )
87eqeq1d 2374 . . . . . . . . 9  |-  ( y  =  <. z ,  z
>.  ->  ( ( 1st `  y )  =  x  <-> 
z  =  x ) )
98pm5.32ri 619 . . . . . . . 8  |-  ( ( ( 1st `  y
)  =  x  /\  y  =  <. z ,  z >. )  <->  ( z  =  x  /\  y  =  <. z ,  z
>. ) )
109exbii 1587 . . . . . . 7  |-  ( E. z ( ( 1st `  y )  =  x  /\  y  =  <. z ,  z >. )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
11 fo1st 6266 . . . . . . . . . 10  |-  1st : _V -onto-> _V
12 fofn 5559 . . . . . . . . . 10  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
1311, 12ax-mp 8 . . . . . . . . 9  |-  1st  Fn  _V
14 fnbrfvb 5670 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  y  e.  _V )  ->  ( ( 1st `  y
)  =  x  <->  y 1st x ) )
1513, 2, 14mp2an 653 . . . . . . . 8  |-  ( ( 1st `  y )  =  x  <->  y 1st x )
16 dfid2 4414 . . . . . . . . . 10  |-  _I  =  { <. z ,  z
>.  |  z  =  z }
1716eleq2i 2430 . . . . . . . . 9  |-  ( y  e.  _I  <->  y  e.  {
<. z ,  z >.  |  z  =  z } )
18 nfe1 1737 . . . . . . . . . . 11  |-  F/ z E. z ( y  =  <. z ,  z
>.  /\  z  =  z )
191819.9 1789 . . . . . . . . . 10  |-  ( E. z E. z ( y  =  <. z ,  z >.  /\  z  =  z )  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
20 elopab 4375 . . . . . . . . . 10  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z E. z ( y  = 
<. z ,  z >.  /\  z  =  z
) )
21 equid 1681 . . . . . . . . . . . 12  |-  z  =  z
2221biantru 491 . . . . . . . . . . 11  |-  ( y  =  <. z ,  z
>. 
<->  ( y  =  <. z ,  z >.  /\  z  =  z ) )
2322exbii 1587 . . . . . . . . . 10  |-  ( E. z  y  =  <. z ,  z >.  <->  E. z
( y  =  <. z ,  z >.  /\  z  =  z ) )
2419, 20, 233bitr4i 268 . . . . . . . . 9  |-  ( y  e.  { <. z ,  z >.  |  z  =  z }  <->  E. z 
y  =  <. z ,  z >. )
2517, 24bitr2i 241 . . . . . . . 8  |-  ( E. z  y  =  <. z ,  z >.  <->  y  e.  _I  )
2615, 25anbi12i 678 . . . . . . 7  |-  ( ( ( 1st `  y
)  =  x  /\  E. z  y  =  <. z ,  z >. )  <->  ( y 1st x  /\  y  e.  _I  )
)
275, 10, 263bitr3ri 267 . . . . . 6  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  E. z ( z  =  x  /\  y  = 
<. z ,  z >.
) )
28 id 19 . . . . . . . . 9  |-  ( z  =  x  ->  z  =  x )
2928, 28opeq12d 3906 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  z >.  =  <. x ,  x >. )
3029eqeq2d 2377 . . . . . . 7  |-  ( z  =  x  ->  (
y  =  <. z ,  z >.  <->  y  =  <. x ,  x >. ) )
311, 30ceqsexv 2908 . . . . . 6  |-  ( E. z ( z  =  x  /\  y  = 
<. z ,  z >.
)  <->  y  =  <. x ,  x >. )
3227, 31bitri 240 . . . . 5  |-  ( ( y 1st x  /\  y  e.  _I  )  <->  y  =  <. x ,  x >. )
334, 32bitri 240 . . . 4  |-  ( y ( 1st  |`  _I  )
x  <->  y  =  <. x ,  x >. )
343, 33bitri 240 . . 3  |-  ( x `' ( 1st  |`  _I  )
y  <->  y  =  <. x ,  x >. )
3534opabbii 4185 . 2  |-  { <. x ,  y >.  |  x `' ( 1st  |`  _I  )
y }  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
36 relcnv 5154 . . 3  |-  Rel  `' ( 1st  |`  _I  )
37 dfrel4v 5228 . . 3  |-  ( Rel  `' ( 1st  |`  _I  )  <->  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y } )
3836, 37mpbi 199 . 2  |-  `' ( 1st  |`  _I  )  =  { <. x ,  y
>.  |  x `' ( 1st  |`  _I  )
y }
39 mptv 4214 . 2  |-  ( x  e.  _V  |->  <. x ,  x >. )  =  { <. x ,  y >.  |  y  =  <. x ,  x >. }
4035, 38, 393eqtr4i 2396 1  |-  `' ( 1st  |`  _I  )  =  ( x  e. 
_V  |->  <. x ,  x >. )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1546    = wceq 1647    e. wcel 1715   _Vcvv 2873   <.cop 3732   class class class wbr 4125   {copab 4178    e. cmpt 4179    _I cid 4407   `'ccnv 4791    |` cres 4794   Rel wrel 4797    Fn wfn 5353   -onto->wfo 5356   ` cfv 5358   1stc1st 6247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-1st 6249
  Copyright terms: Public domain W3C validator