Theorem dfrel2 5061

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 4989	. . 3 ⊢ Rel ◡◡𝑅
2		vex 2733	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2733	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelcnv 4793	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
5	3, 2	opelcnv 4793	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6	4, 5	bitri 183	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7	6	eqrelriv 4704	. . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅)
8	1, 7	mpan 422	. 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9		releq 4693	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅))
10	1, 9	mpbii 147	. 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅)
11	8, 10	impbii 125	1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)

Syntax hints:   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ⟨cop 3586  ^◡ccnv 4610  Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619

This theorem is referenced by:  dfrel4v  5062  cnvcnv  5063  cnveqb  5066  dfrel3  5068  cnvcnvres  5074  cnvsn  5093  cores2  5123  co01  5125  coi2  5127  relcnvtr  5130  relcnvexb  5150  funcnvres2  5273  f1cnvcnv  5414  f1ocnv  5455  f1ocnvb  5456  f1ococnv1  5471  isores1  5793  cnvf1o  6204  tposf12  6248  ssenen  6829  relcnvfi  6918  caseinl  7068  caseinr  7069  fsumcnv  11400  fprodcnv  11588  structcnvcnv  12432  hmeocnv  13101  hmeocnvb  13112