Theorem dfrel2 2660

Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25.

Assertion

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

Proof of Theorem dfrel2

Step	Hyp	Ref	Expression
1		relcnv 2624	. . 3 ⊢ Rel ◡◡R
2		visset 1350	. . . . . . 7 ⊢ x ∈ V
3		visset 1350	. . . . . . 7 ⊢ y ∈ V
4	2, 3	opelcnv 2518	. . . . . 6 ⊢ (⟨x, y⟩ ∈ ◡◡R ↔ ⟨y, x⟩ ∈ ◡R)
5	3, 2	opelcnv 2518	. . . . . 6 ⊢ (⟨y, x⟩ ∈ ◡R ↔ ⟨x, y⟩ ∈ R)
6	4, 5	bitr 151	. . . . 5 ⊢ (⟨x, y⟩ ∈ ◡◡R ↔ ⟨x, y⟩ ∈ R)
7	6	gen2 681	. . . 4 ⊢ ∀x∀y(⟨x, y⟩ ∈ ◡◡R ↔ ⟨x, y⟩ ∈ R)
8		cleqrel 2483	. . . 4 ⊢ ((Rel ◡◡R ∧ Rel R) → (◡◡R = R ↔ ∀x∀y(⟨x, y⟩ ∈ ◡◡R ↔ ⟨x, y⟩ ∈ R)))
9	7, 8	mpbiri 169	. . 3 ⊢ ((Rel ◡◡R ∧ Rel R) → ◡◡R = R)
10	1, 9	mpan 518	. 2 ⊢ (Rel R → ◡◡R = R)
11		releq 2477	. . 3 ⊢ (◡◡R = R → (Rel ◡◡R ↔ Rel R))
12	1, 11	mpbii 168	. 2 ⊢ (◡◡R = R → Rel R)
13	10, 12	impbi 139	1 ⊢ (Rel R ↔ ◡◡R = R)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  ^◡ccnv 2409  Rel wrel 2415

This theorem is referenced by:  cnvcnv 2661  co01 2664  coi2 2666  funimacnv 2711  f1cnv 2782  f1ocnvb 2812  f1ococnv1 2818  ssenen 3399

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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