Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ofmres Structured version   Unicode version

Theorem ofmres 5705
 Description: Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, 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 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 2959 . . 3
2 ssv 2959 . . 3
3 resmpt2 5541 . . 3
41, 2, 3mp2an 402 . 2
5 df-of 5654 . . 3
65reseq1i 4551 . 2
7 eqid 2037 . . 3
8 eqid 2037 . . 3
9 vex 2554 . . . 4
10 vex 2554 . . . 4
119dmex 4541 . . . . . 6
1211inex1 3882 . . . . 5
1312mptex 5330 . . . 4
145ovmpt4g 5565 . . . 4
159, 10, 13, 14mp3an 1231 . . 3
167, 8, 15mpt2eq123i 5510 . 2
174, 6, 163eqtr4i 2067 1
 Colors of variables: wff set class Syntax hints:   wceq 1242   wcel 1390  cvv 2551   cin 2910   wss 2911   cmpt 3809   cxp 4286   cdm 4288   cres 4290  cfv 4845  (class class class)co 5455   cmpt2 5457   cof 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)
 Copyright terms: Public domain W3C validator