Theorem cnvsym 5050

Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvsym	⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsym

Step	Hyp	Ref	Expression
1		alcom 1489	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2		relcnv 5044	. . 3 ⊢ Rel ◡𝑅
3		ssrel 4748	. . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
4	2, 3	ax-mp 5	. 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5		vex 2763	. . . . . 6 ⊢ 𝑦 ∈ V
6		vex 2763	. . . . . 6 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4846	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8		df-br 4031	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 186	. . . 4 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
10		df-br 4031	. . . 4 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
11	9, 10	imbi12i 239	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12	11	2albii 1482	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
13	1, 4, 12	3bitr4i 212	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   ∈ wcel 2164   ⊆ wss 3154  ⟨cop 3622   class class class wbr 4030  ^◡ccnv 4659  Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668

This theorem is referenced by:  dfer2  6590