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

Theorem ofmres 6803
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 6804, allowing it to be used as a function or structure argument. By ofmresval 6558, 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 3490 . . 3  |-  A  C_  _V
2 ssv 3490 . . 3  |-  B  C_  _V
3 resmpt2 6408 . . 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 676 . 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 6545 . . 3  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
65reseq1i 5121 . 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 2429 . . 3  |-  A  =  A
8 eqid 2429 . . 3  |-  B  =  B
9 vex 3090 . . . 4  |-  f  e. 
_V
10 vex 3090 . . . 4  |-  g  e. 
_V
119dmex 6740 . . . . . 6  |-  dom  f  e.  _V
1211inex1 4566 . . . . 5  |-  ( dom  f  i^i  dom  g
)  e.  _V
1312mptex 6151 . . . 4  |-  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )  e.  _V
145ovmpt4g 6433 . . . 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 1360 . . 3  |-  ( f  oF R g )  =  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )
167, 8, 15mpt2eq123i 6368 . 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 2468 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 1437    e. wcel 1870   _Vcvv 3087    i^i cin 3441    C_ wss 3442    |-> cmpt 4484    X. cxp 4852   dom cdm 4854    |` cres 4856   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307    oFcof 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545
This theorem is referenced by:  mplsubrglem  18598  psrplusgpropd  18764
  Copyright terms: Public domain W3C validator