ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfrel3 GIF version

Theorem dfrel3 4806
Description: Alternate definition of relation. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
dfrel3 (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Proof of Theorem dfrel3
StepHypRef Expression
1 dfrel2 4799 . 2 (Rel 𝑅𝑅 = 𝑅)
2 cnvcnv2 4802 . . 3 𝑅 = (𝑅 ↾ V)
32eqeq1i 2063 . 2 (𝑅 = 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
41, 3bitri 177 1 (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  Vcvv 2574  ccnv 4372  cres 4375  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-res 4385
This theorem is referenced by:  cocnvcnv2  4860  f1ovi  5193
  Copyright terms: Public domain W3C validator