Theorem cnvsym 4987

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 1466	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2		relcnv 4982	. . 3 ⊢ Rel ◡𝑅
3		ssrel 4692	. . 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 2729	. . . . . 6 ⊢ 𝑦 ∈ V
6		vex 2729	. . . . . 6 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4787	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8		df-br 3983	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 185	. . . 4 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
10		df-br 3983	. . . 4 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
11	9, 10	imbi12i 238	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12	11	2albii 1459	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
13	1, 4, 12	3bitr4i 211	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341   ∈ wcel 2136   ⊆ wss 3116  ⟨cop 3579   class class class wbr 3982  ^◡ccnv 4603  Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  dfer2  6502