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Theorem ofmres 6118
Description: Equivalent expressions for a restriction of the function operation map. Unlike  o F R which is a proper class,  (  o F R  |  `  ( A  X.  B
) ) can be a set by ofmresex 6120, allowing it to be used as a function or structure argument. By ofmresval 6119, the restricted operation map values are the same as the original values, allowing theorems for  o F R to be reused. (Contributed by NM, 20-Oct-2014.)
Assertion
Ref Expression
ofmres  |-  (  o F R  |`  ( A  X.  B ) )  =  ( f  e.  A ,  g  e.  B  |->  ( f  o F 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 3200 . . 3  |-  A  C_  _V
2 ssv 3200 . . 3  |-  B  C_  _V
3 resmpt2 5944 . . 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 653 . 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 6080 . . 3  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
65reseq1i 4953 . 2  |-  (  o F 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 2285 . . 3  |-  A  =  A
8 eqid 2285 . . 3  |-  B  =  B
9 vex 2793 . . . 4  |-  f  e. 
_V
10 vex 2793 . . . 4  |-  g  e. 
_V
119dmex 4943 . . . . . 6  |-  dom  f  e.  _V
1211inex1 4157 . . . . 5  |-  ( dom  f  i^i  dom  g
)  e.  _V
1312mptex 5748 . . . 4  |-  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )  e.  _V
145ovmpt4g 5972 . . . 4  |-  ( ( f  e.  _V  /\  g  e.  _V  /\  (
x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) )  e. 
_V )  ->  (
f  o F R g )  =  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )
159, 10, 13, 14mp3an 1277 . . 3  |-  ( f  o F R g )  =  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )
167, 8, 15mpt2eq123i 5913 . 2  |-  ( f  e.  A ,  g  e.  B  |->  ( f  o F R g ) )  =  ( f  e.  A , 
g  e.  B  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )
174, 6, 163eqtr4i 2315 1  |-  (  o F R  |`  ( A  X.  B ) )  =  ( f  e.  A ,  g  e.  B  |->  ( f  o F R g ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1625    e. wcel 1686   _Vcvv 2790    i^i cin 3153    C_ wss 3154    e. cmpt 4079    X. cxp 4689   dom cdm 4691    |` cres 4693   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862    o Fcof 6078
This theorem is referenced by:  mplsubrglem  16185  psrplusgpropd  16315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080
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