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Theorem dfrel2 4984
Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dfrel2 (Rel 𝑅𝑅 = 𝑅)

Proof of Theorem dfrel2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4912 . . 3 Rel 𝑅
2 vex 2684 . . . . . 6 𝑥 ∈ V
3 vex 2684 . . . . . 6 𝑦 ∈ V
42, 3opelcnv 4716 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
53, 2opelcnv 4716 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
64, 5bitri 183 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
76eqrelriv 4627 . . 3 ((Rel 𝑅 ∧ Rel 𝑅) → 𝑅 = 𝑅)
81, 7mpan 420 . 2 (Rel 𝑅𝑅 = 𝑅)
9 releq 4616 . . 3 (𝑅 = 𝑅 → (Rel 𝑅 ↔ Rel 𝑅))
101, 9mpbii 147 . 2 (𝑅 = 𝑅 → Rel 𝑅)
118, 10impbii 125 1 (Rel 𝑅𝑅 = 𝑅)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1331  wcel 1480  cop 3525  ccnv 4533  Rel wrel 4539
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542
This theorem is referenced by:  dfrel4v  4985  cnvcnv  4986  cnveqb  4989  dfrel3  4991  cnvcnvres  4997  cnvsn  5016  cores2  5046  co01  5048  coi2  5050  relcnvtr  5053  relcnvexb  5073  funcnvres2  5193  f1cnvcnv  5334  f1ocnv  5373  f1ocnvb  5374  f1ococnv1  5389  isores1  5708  cnvf1o  6115  tposf12  6159  ssenen  6738  relcnvfi  6822  caseinl  6969  caseinr  6970  fsumcnv  11199  structcnvcnv  11964  hmeocnv  12465  hmeocnvb  12476
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