Theorem dfrel2 5212

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 5139	. . 3 ⊢ Rel ◡◡𝑅
2		vex 2815	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2815	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelcnv 4936	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
5	3, 2	opelcnv 4936	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6	4, 5	bitri 184	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7	6	eqrelriv 4842	. . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅)
8	1, 7	mpan 424	. 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9		releq 4831	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅))
10	1, 9	mpbii 148	. 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅)
11	8, 10	impbii 126	1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ⟨cop 3691  ^◡ccnv 4747  Rel wrel 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756

This theorem is referenced by:  dfrel4v  5213  cnvcnv  5214  cnveqb  5217  dfrel3  5219  cnvcnvres  5225  cnvsn  5244  cores2  5274  co01  5276  coi2  5278  relcnvtr  5281  relcnvexb  5301  funcnvres2  5430  f1cnvcnv  5583  f1ocnv  5626  f1ocnvb  5627  f1ococnv1  5642  isores1  5986  cnvf1o  6420  tposf12  6499  ssenen  7104  relcnvfi  7207  caseinl  7381  caseinr  7382  fsumcnv  12119  fprodcnv  12307  structcnvcnv  13220  hmeocnv  15164  hmeocnvb  15175