Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffn5im Unicode version

Theorem dffn5im 5219
 Description: Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 4938 and dmmptss 4817. (Contributed by Jim Kingdon, 31-Dec-2018.)
Assertion
Ref Expression
dffn5im
Distinct variable groups:   ,   ,

Proof of Theorem dffn5im
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fnrel 4997 . . . 4
2 dfrel4v 4772 . . . 4
31, 2sylib 127 . . 3
4 fnbr 5001 . . . . . . 7
54ex 108 . . . . . 6
65pm4.71rd 374 . . . . 5
7 eqcom 2042 . . . . . . 7
8 fnbrfvb 5214 . . . . . . 7
97, 8syl5bb 181 . . . . . 6
109pm5.32da 425 . . . . 5
116, 10bitr4d 180 . . . 4
1211opabbidv 3823 . . 3
133, 12eqtrd 2072 . 2
14 df-mpt 3820 . 2
1513, 14syl6eqr 2090 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wceq 1243   wcel 1393   class class class wbr 3764  copab 3817   cmpt 3818   wrel 4350   wfn 4897  cfv 4902 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  fnrnfv  5220  feqmptd  5226  dffn5imf  5228  eqfnfv  5265  fndmin  5274  fcompt  5333  resfunexg  5382  eufnfv  5389  fnovim  5609  offveqb  5730  caofinvl  5733  oprabco  5838  df1st2  5840  df2nd2  5841
 Copyright terms: Public domain W3C validator