Theorem cnvsym 5166

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 1531	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2		relcnv 5160	. . 3 ⊢ Rel ◡𝑅
3		ssrel 4858	. . 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 2824	. . . . . 6 ⊢ 𝑦 ∈ V
6		vex 2824	. . . . . 6 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4958	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8		df-br 4126	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 186	. . . 4 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
10		df-br 4126	. . . 4 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
11	9, 10	imbi12i 239	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12	11	2albii 1524	. 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 1400   ∈ wcel 2209   ⊆ wss 3220  ⟨cop 3708   class class class wbr 4125  ^◡ccnv 4768  Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777

This theorem is referenced by:  dfer2  6798