Theorem dfrel2 5182

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 5109	. . 3 ⊢ Rel ◡◡𝑅
2		vex 2802	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2802	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelcnv 4907	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
5	3, 2	opelcnv 4907	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6	4, 5	bitri 184	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7	6	eqrelriv 4814	. . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅)
8	1, 7	mpan 424	. 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9		releq 4803	. . 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 1395   ∈ wcel 2200  ⟨cop 3669  ^◡ccnv 4719  Rel wrel 4725

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4726  df-rel 4727  df-cnv 4728

This theorem is referenced by:  dfrel4v  5183  cnvcnv  5184  cnveqb  5187  dfrel3  5189  cnvcnvres  5195  cnvsn  5214  cores2  5244  co01  5246  coi2  5248  relcnvtr  5251  relcnvexb  5271  funcnvres2  5399  f1cnvcnv  5547  f1ocnv  5590  f1ocnvb  5591  f1ococnv1  5606  isores1  5947  cnvf1o  6382  tposf12  6426  ssenen  7025  relcnvfi  7124  caseinl  7274  caseinr  7275  fsumcnv  11969  fprodcnv  12157  structcnvcnv  13069  hmeocnv  15002  hmeocnvb  15013