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Theorem ofmres 6780
 Description: Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6781, allowing it to be used as a function or structure argument. By ofmresval 6536, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)
Assertion
Ref Expression
ofmres
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem ofmres
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssv 3524 . . 3
2 ssv 3524 . . 3
3 resmpt2 6384 . . 3
41, 2, 3mp2an 672 . 2
5 df-of 6524 . . 3
65reseq1i 5269 . 2
7 eqid 2467 . . 3
8 eqid 2467 . . 3
9 vex 3116 . . . 4
10 vex 3116 . . . 4
119dmex 6717 . . . . . 6
1211inex1 4588 . . . . 5
1312mptex 6131 . . . 4
145ovmpt4g 6409 . . . 4
159, 10, 13, 14mp3an 1324 . . 3
167, 8, 15mpt2eq123i 6344 . 2
174, 6, 163eqtr4i 2506 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1379   wcel 1767  cvv 3113   cin 3475   wss 3476   cmpt 4505   cxp 4997   cdm 4999   cres 5001  cfv 5588  (class class class)co 6284   cmpt2 6286   cof 6522 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524 This theorem is referenced by:  mplsubrglem  17899  mplsubrglemOLD  17900  psrplusgpropd  18076
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