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Theorem ofmres 6682
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 6683, allowing it to be used as a function or structure argument. By ofmresval 6441, 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 3483 . . 3  |-  A  C_  _V
2 ssv 3483 . . 3  |-  B  C_  _V
3 resmpt2 6297 . . 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 6429 . . 3  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
65reseq1i 5213 . 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 2454 . . 3  |-  A  =  A
8 eqid 2454 . . 3  |-  B  =  B
9 vex 3079 . . . 4  |-  f  e. 
_V
10 vex 3079 . . . 4  |-  g  e. 
_V
119dmex 6620 . . . . . 6  |-  dom  f  e.  _V
1211inex1 4540 . . . . 5  |-  ( dom  f  i^i  dom  g
)  e.  _V
1312mptex 6056 . . . 4  |-  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )  e.  _V
145ovmpt4g 6322 . . . 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 1315 . . 3  |-  ( f  oF R g )  =  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )
167, 8, 15mpt2eq123i 6257 . 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 2493 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 1370    e. wcel 1758   _Vcvv 3076    i^i cin 3434    C_ wss 3435    |-> cmpt 4457    X. cxp 4945   dom cdm 4947    |` cres 4949   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201    oFcof 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429
This theorem is referenced by:  mplsubrglem  17640  mplsubrglemOLD  17641  psrplusgpropd  17813
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