Theorem cnvsym 4651

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 1364	. 2 ⊢ (∀y∀x(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅) ↔ ∀x∀y(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅))
2		relcnv 4646	. . 3 ⊢ Rel ◡𝑅
3		ssrel 4371	. . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀y∀x(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅)))
4	2, 3	ax-mp 7	. 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀y∀x(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅))
5		vex 2554	. . . . . 6 ⊢ y ∈ V
6		vex 2554	. . . . . 6 ⊢ x ∈ V
7	5, 6	brcnv 4461	. . . . 5 ⊢ (y◡𝑅x ↔ x𝑅y)
8		df-br 3756	. . . . 5 ⊢ (y◡𝑅x ↔ ⟨y, x⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 175	. . . 4 ⊢ (x𝑅y ↔ ⟨y, x⟩ ∈ ◡𝑅)
10		df-br 3756	. . . 4 ⊢ (y𝑅x ↔ ⟨y, x⟩ ∈ 𝑅)
11	9, 10	imbi12i 228	. . 3 ⊢ ((x𝑅y → y𝑅x) ↔ (⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅))
12	11	2albii 1357	. 2 ⊢ (∀x∀y(x𝑅y → y𝑅x) ↔ ∀x∀y(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅))
13	1, 4, 12	3bitr4i 201	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀x∀y(x𝑅y → y𝑅x))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  ^◡ccnv 4287  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296

This theorem is referenced by:  dfer2  6043