Theorem dfrel2 4881

Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Assertion

Ref	Expression
dfrel2	⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)

Proof of Theorem dfrel2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 4810	. . 3 ⊢ Rel ◡◡𝑅
2		vex 2622	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2622	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelcnv 4618	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
5	3, 2	opelcnv 4618	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6	4, 5	bitri 182	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7	6	eqrelriv 4531	. . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅)
8	1, 7	mpan 415	. 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9		releq 4520	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅))
10	1, 9	mpbii 146	. 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅)
11	8, 10	impbii 124	1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)

Syntax hints:   ↔ wb 103   = wceq 1289   ∈ wcel 1438  ⟨cop 3449  ^◡ccnv 4437  Rel wrel 4443

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446

This theorem is referenced by:  dfrel4v  4882  cnvcnv  4883  cnveqb  4886  dfrel3  4888  cnvcnvres  4894  cnvsn  4913  cores2  4943  co01  4945  coi2  4947  relcnvtr  4950  relcnvexb  4970  funcnvres2  5089  f1cnvcnv  5227  f1ocnv  5266  f1ocnvb  5267  f1ococnv1  5282  isores1  5593  cnvf1o  5990  tposf12  6034  ssenen  6565  relcnvfi  6648  fsumcnv  10827  structcnvcnv  11506