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Theorem ofmres 6074
 Description: Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6075, allowing it to be used as a function or structure argument. By ofmresval 6033, 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 3146 . . 3
2 ssv 3146 . . 3
3 resmpo 5909 . . 3
41, 2, 3mp2an 423 . 2
5 df-of 6022 . . 3
65reseq1i 4855 . 2
7 eqid 2154 . . 3
8 eqid 2154 . . 3
9 vex 2712 . . . 4
10 vex 2712 . . . 4
119dmex 4845 . . . . . 6
1211inex1 4094 . . . . 5
1312mptex 5686 . . . 4
145ovmpt4g 5933 . . . 4
159, 10, 13, 14mp3an 1316 . . 3
167, 8, 15mpoeq123i 5874 . 2
174, 6, 163eqtr4i 2185 1
 Colors of variables: wff set class Syntax hints:   wceq 1332   wcel 2125  cvv 2709   cin 3097   wss 3098   cmpt 4021   cxp 4577   cdm 4579   cres 4581  cfv 5163  (class class class)co 5814   cmpo 5816   cof 6020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-of 6022 This theorem is referenced by: (None)
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