Theorem dfrel2 4714

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 4646	. . 3 ⊢ Rel ◡◡𝑅
2		vex 2554	. . . . . 6 ⊢ x ∈ V
3		vex 2554	. . . . . 6 ⊢ y ∈ V
4	2, 3	opelcnv 4460	. . . . 5 ⊢ (⟨x, y⟩ ∈ ◡◡𝑅 ↔ ⟨y, x⟩ ∈ ◡𝑅)
5	3, 2	opelcnv 4460	. . . . 5 ⊢ (⟨y, x⟩ ∈ ◡𝑅 ↔ ⟨x, y⟩ ∈ 𝑅)
6	4, 5	bitri 173	. . . 4 ⊢ (⟨x, y⟩ ∈ ◡◡𝑅 ↔ ⟨x, y⟩ ∈ 𝑅)
7	6	eqrelriv 4376	. . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅)
8	1, 7	mpan 400	. 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9		releq 4365	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅))
10	1, 9	mpbii 136	. 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅)
11	8, 10	impbii 117	1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)

Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ⟨cop 3370  ^◡ccnv 4287  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296

This theorem is referenced by:  dfrel4v  4715  cnvcnv  4716  cnveqb  4719  dfrel3  4721  cnvcnvres  4727  cnvsn  4746  cores2  4776  co01  4778  coi2  4780  relcnvtr  4783  relcnvexb  4800  funcnvres2  4917  f1cnvcnv  5043  f1ocnv  5082  f1ocnvb  5083  f1ococnv1  5098  isores1  5397  cnvf1o  5788  tposf12  5825