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Theorem dfrel2 4798
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 4730 . . 3 Rel 𝑅
2 vex 2577 . . . . . 6 𝑥 ∈ V
3 vex 2577 . . . . . 6 𝑦 ∈ V
42, 3opelcnv 4544 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
53, 2opelcnv 4544 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
64, 5bitri 177 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
76eqrelriv 4460 . . 3 ((Rel 𝑅 ∧ Rel 𝑅) → 𝑅 = 𝑅)
81, 7mpan 408 . 2 (Rel 𝑅𝑅 = 𝑅)
9 releq 4449 . . 3 (𝑅 = 𝑅 → (Rel 𝑅 ↔ Rel 𝑅))
101, 9mpbii 140 . 2 (𝑅 = 𝑅 → Rel 𝑅)
118, 10impbii 121 1 (Rel 𝑅𝑅 = 𝑅)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wcel 1409  cop 3405  ccnv 4371  Rel wrel 4377
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 3902  ax-pow 3954  ax-pr 3971
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 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380
This theorem is referenced by:  dfrel4v  4799  cnvcnv  4800  cnveqb  4803  dfrel3  4805  cnvcnvres  4811  cnvsn  4830  cores2  4860  co01  4862  coi2  4864  relcnvtr  4867  relcnvexb  4884  funcnvres2  5001  f1cnvcnv  5127  f1ocnv  5166  f1ocnvb  5167  f1ococnv1  5182  isores1  5481  cnvf1o  5873  tposf12  5914
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