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Theorem dfrel2 5502
 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 5422 . . 3 Rel 𝑅
2 vex 3176 . . . . . 6 𝑥 ∈ V
3 vex 3176 . . . . . 6 𝑦 ∈ V
42, 3opelcnv 5226 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
53, 2opelcnv 5226 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
64, 5bitri 263 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
76eqrelriv 5136 . . 3 ((Rel 𝑅 ∧ Rel 𝑅) → 𝑅 = 𝑅)
81, 7mpan 702 . 2 (Rel 𝑅𝑅 = 𝑅)
9 releq 5124 . . 3 (𝑅 = 𝑅 → (Rel 𝑅 ↔ Rel 𝑅))
101, 9mpbii 222 . 2 (𝑅 = 𝑅 → Rel 𝑅)
118, 10impbii 198 1 (Rel 𝑅𝑅 = 𝑅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ◡ccnv 5037  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  dfrel4v  5503  cnvcnv  5505  cnveqb  5508  dfrel3  5510  cnvcnvres  5516  cnvsn  5536  cores2  5565  co01  5567  coi2  5569  relcnvtr  5572  funcnvres2  5883  f1cnvcnv  6022  f1ocnv  6062  f1ocnvb  6063  f1ococnv1  6078  fimacnvinrn  6256  isores1  6484  relcnvexb  7007  cnvf1o  7163  fnwelem  7179  tposf12  7264  ssenen  8019  cantnffval2  8475  fsumcnv  14346  fprodcnv  14552  structcnvcnv  15706  imasless  16023  oppcinv  16263  cnvps  17035  cnvpsb  17036  cnvtsr  17045  gimcnv  17532  lmimcnv  18888  hmeocnv  21375  hmeocnvb  21387  cmphaushmeo  21413  ustexsym  21829  pi1xfrcnv  22665  dvlog  24197  efopnlem2  24203  gtiso  28861  f1ocan2fv  32692  ltrncnvnid  34431  relintab  36908  cnvssb  36911  relnonrel  36912  cononrel1  36919  cononrel2  36920  clrellem  36948  clcnvlem  36949  relexpaddss  37029
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