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

Theorem ofmres 6780
Description: Equivalent expressions for a restriction of the function operation map. Unlike  oF R which is a proper class,  (  oF R  |  `  ( A  X.  B
) ) can be a set by ofmresex 6781, allowing it to be used as a function or structure argument. By ofmresval 6536, the restricted operation map values are the same as the original values, allowing theorems for  oF R to be reused. (Contributed by NM, 20-Oct-2014.)
Assertion
Ref Expression
ofmres  |-  (  oF R  |`  ( A  X.  B ) )  =  ( f  e.  A ,  g  e.  B  |->  ( f  oF R g ) )
Distinct variable groups:    f, g, A    B, f, g    R, f, g

Proof of Theorem ofmres
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssv 3524 . . 3  |-  A  C_  _V
2 ssv 3524 . . 3  |-  B  C_  _V
3 resmpt2 6384 . . 3  |-  ( ( A  C_  _V  /\  B  C_ 
_V )  ->  (
( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )  |`  ( A  X.  B
) )  =  ( f  e.  A , 
g  e.  B  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) ) )
41, 2, 3mp2an 672 . 2  |-  ( ( f  e.  _V , 
g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )  |`  ( A  X.  B
) )  =  ( f  e.  A , 
g  e.  B  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )
5 df-of 6524 . . 3  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
65reseq1i 5269 . 2  |-  (  oF R  |`  ( A  X.  B ) )  =  ( ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )  |`  ( A  X.  B ) )
7 eqid 2467 . . 3  |-  A  =  A
8 eqid 2467 . . 3  |-  B  =  B
9 vex 3116 . . . 4  |-  f  e. 
_V
10 vex 3116 . . . 4  |-  g  e. 
_V
119dmex 6717 . . . . . 6  |-  dom  f  e.  _V
1211inex1 4588 . . . . 5  |-  ( dom  f  i^i  dom  g
)  e.  _V
1312mptex 6131 . . . 4  |-  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )  e.  _V
145ovmpt4g 6409 . . . 4  |-  ( ( f  e.  _V  /\  g  e.  _V  /\  (
x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) )  e. 
_V )  ->  (
f  oF R g )  =  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )
159, 10, 13, 14mp3an 1324 . . 3  |-  ( f  oF R g )  =  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )
167, 8, 15mpt2eq123i 6344 . 2  |-  ( f  e.  A ,  g  e.  B  |->  ( f  oF R g ) )  =  ( f  e.  A , 
g  e.  B  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )
174, 6, 163eqtr4i 2506 1  |-  (  oF R  |`  ( A  X.  B ) )  =  ( f  e.  A ,  g  e.  B  |->  ( f  oF R g ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476    |-> cmpt 4505    X. cxp 4997   dom cdm 4999    |` cres 5001   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286    oFcof 6522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524
This theorem is referenced by:  mplsubrglem  17899  mplsubrglemOLD  17900  psrplusgpropd  18076
  Copyright terms: Public domain W3C validator