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Theorem ofmres 5705
Description: Equivalent expressions for a restriction of the function operation map. Unlike  o F R which is a proper class,  o F R  |  `  X. can be a set by ofmresex 5706, allowing it to be used as a function or structure argument. By ofmresval 5665, 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  |`  X.  ,  |->  o F R
Distinct variable groups:   ,,   ,,    R,,

Proof of Theorem ofmres
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssv 2959 . . 3  C_  _V
2 ssv 2959 . . 3  C_  _V
3 resmpt2 5541 . . 3  C_  _V  C_ 
_V  _V ,  _V  |->  dom  i^i  dom  |->  `
 R `  |`  X.  ,  |->  dom  i^i  dom  |->  `  R `
41, 2, 3mp2an 402 . 2  _V ,  _V  |->  dom  i^i  dom  |->  `  R `  |`  X.  ,  |->  dom  i^i  dom  |->  `  R `
5 df-of 5654 . . 3  o F R  _V ,  _V  |->  dom  i^i  dom  |->  `  R `
65reseq1i 4551 . 2  o F R  |`  X.  _V ,  _V  |->  dom  i^i  dom  |->  `  R `  |`  X.
7 eqid 2037 . . 3
8 eqid 2037 . . 3
9 vex 2554 . . . 4 
_V
10 vex 2554 . . . 4 
_V
119dmex 4541 . . . . . 6  dom  _V
1211inex1 3882 . . . . 5  dom  i^i  dom  _V
1312mptex 5330 . . . 4  dom  i^i  dom  |->  `  R ` 
_V
145ovmpt4g 5565 . . . 4  _V  _V  dom  i^i  dom  |->  `  R `
 _V  o F R  dom  i^i  dom  |->  `  R `
159, 10, 13, 14mp3an 1231 . . 3  o F R  dom  i^i  dom  |->  `  R `
167, 8, 15mpt2eq123i 5510 . 2  ,  |->  o F R  ,  |->  dom  i^i  dom  |->  `  R `
174, 6, 163eqtr4i 2067 1  o F R  |`  X.  ,  |->  o F R
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390   _Vcvv 2551    i^i cin 2910    C_ wss 2911    |-> cmpt 3809    X. cxp 4286   dom cdm 4288    |` cres 4290   ` cfv 4845  (class class class)co 5455    |-> cmpt2 5457    o Fcof 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654
This theorem is referenced by: (None)
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