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Theorem dfrel2 5081
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 5008 . . 3 Rel 𝑅
2 vex 2742 . . . . . 6 𝑥 ∈ V
3 vex 2742 . . . . . 6 𝑦 ∈ V
42, 3opelcnv 4811 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
53, 2opelcnv 4811 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
64, 5bitri 184 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
76eqrelriv 4721 . . 3 ((Rel 𝑅 ∧ Rel 𝑅) → 𝑅 = 𝑅)
81, 7mpan 424 . 2 (Rel 𝑅𝑅 = 𝑅)
9 releq 4710 . . 3 (𝑅 = 𝑅 → (Rel 𝑅 ↔ Rel 𝑅))
101, 9mpbii 148 . 2 (𝑅 = 𝑅 → Rel 𝑅)
118, 10impbii 126 1 (Rel 𝑅𝑅 = 𝑅)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  cop 3597  ccnv 4627  Rel wrel 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636
This theorem is referenced by:  dfrel4v  5082  cnvcnv  5083  cnveqb  5086  dfrel3  5088  cnvcnvres  5094  cnvsn  5113  cores2  5143  co01  5145  coi2  5147  relcnvtr  5150  relcnvexb  5170  funcnvres2  5293  f1cnvcnv  5434  f1ocnv  5476  f1ocnvb  5477  f1ococnv1  5492  isores1  5817  cnvf1o  6228  tposf12  6272  ssenen  6853  relcnvfi  6942  caseinl  7092  caseinr  7093  fsumcnv  11447  fprodcnv  11635  structcnvcnv  12480  hmeocnv  13846  hmeocnvb  13857
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