Theorem cnvsym 4631

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 1343	. 2 ⊢ (∀y∀x(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅) ↔ ∀x∀y(⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅))
2		relcnv 4626	. . 3 ⊢ Rel ◡𝑅
3		ssrel 4351	. . 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 2534	. . . . . 6 ⊢ y ∈ V
6		vex 2534	. . . . . 6 ⊢ x ∈ V
7	5, 6	brcnv 4441	. . . . 5 ⊢ (y◡𝑅x ↔ x𝑅y)
8		df-br 3735	. . . . 5 ⊢ (y◡𝑅x ↔ ⟨y, x⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 175	. . . 4 ⊢ (x𝑅y ↔ ⟨y, x⟩ ∈ ◡𝑅)
10		df-br 3735	. . . 4 ⊢ (y𝑅x ↔ ⟨y, x⟩ ∈ 𝑅)
11	9, 10	imbi12i 228	. . 3 ⊢ ((x𝑅y → y𝑅x) ↔ (⟨y, x⟩ ∈ ◡𝑅 → ⟨y, x⟩ ∈ 𝑅))
12	11	2albii 1336	. 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 1224   ∈ wcel 1370   ⊆ wss 2890  ⟨cop 3349   class class class wbr 3734  ^◡ccnv 4267  Rel wrel 4273

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276

This theorem is referenced by:  dfer2  6014