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Theorem ofmres 6203
 Description: Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6205, allowing it to be used as a function or structure argument. By ofmresval 6204, 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 3274 . . 3
2 ssv 3274 . . 3
3 resmpt2 6029 . . 3
41, 2, 3mp2an 653 . 2
5 df-of 6165 . . 3
65reseq1i 5033 . 2
7 eqid 2358 . . 3
8 eqid 2358 . . 3
9 vex 2867 . . . 4
10 vex 2867 . . . 4
119dmex 5023 . . . . . 6
1211inex1 4236 . . . . 5
1312mptex 5832 . . . 4
145ovmpt4g 6057 . . . 4
159, 10, 13, 14mp3an 1277 . . 3
167, 8, 15mpt2eq123i 5998 . 2
174, 6, 163eqtr4i 2388 1
 Colors of variables: wff set class Syntax hints:   wceq 1642   wcel 1710  cvv 2864   cin 3227   wss 3228   cmpt 4158   cxp 4769   cdm 4771   cres 4773  cfv 5337  (class class class)co 5945   cmpt2 5947   cof 6163 This theorem is referenced by:  mplsubrglem  16282  psrplusgpropd  16412 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165
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