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Theorem ofmres 6345
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 6347, allowing it to be used as a function or structure argument. By ofmresval 6346, 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 3370 . . 3  |-  A  C_  _V
2 ssv 3370 . . 3  |-  B  C_  _V
3 resmpt2 6170 . . 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 655 . 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 6307 . . 3  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
65reseq1i 5144 . 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 2438 . . 3  |-  A  =  A
8 eqid 2438 . . 3  |-  B  =  B
9 vex 2961 . . . 4  |-  f  e. 
_V
10 vex 2961 . . . 4  |-  g  e. 
_V
119dmex 5134 . . . . . 6  |-  dom  f  e.  _V
1211inex1 4346 . . . . 5  |-  ( dom  f  i^i  dom  g
)  e.  _V
1312mptex 5968 . . . 4  |-  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )  e.  _V
145ovmpt4g 6198 . . . 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 1280 . . 3  |-  ( f  o F R g )  =  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )
167, 8, 15mpt2eq123i 6139 . 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 2468 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 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322    e. cmpt 4268    X. cxp 4878   dom cdm 4880    |` cres 4882   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    o Fcof 6305
This theorem is referenced by:  mplsubrglem  16504  psrplusgpropd  16631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307
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