Theorem cnvsym 2626

Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51.

Assertion

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

Distinct variable group(s):   x,y,R

Proof of Theorem cnvsym

Step	Hyp	Ref	Expression
1		df-cnv 2426	. . . . 5 ⊢ ◡R = {⟨y, x⟩∣xRy}
2	1	sseq1i 1524	. . . 4 ⊢ (◡R ⊆ R ↔ {⟨y, x⟩∣xRy} ⊆ R)
3		ssel 1502	. . . . . 6 ⊢ ({⟨y, x⟩∣xRy} ⊆ R → (⟨y, x⟩ ∈ {⟨y, x⟩∣xRy} → ⟨y, x⟩ ∈ R))
4		df-br 2063	. . . . . 6 ⊢ (yRx ↔ ⟨y, x⟩ ∈ R)
5	3, 4	syl6ibr 186	. . . . 5 ⊢ ({⟨y, x⟩∣xRy} ⊆ R → (⟨y, x⟩ ∈ {⟨y, x⟩∣xRy} → yRx))
6		opabid 2099	. . . . 5 ⊢ (⟨y, x⟩ ∈ {⟨y, x⟩∣xRy} ↔ xRy)
7	5, 6	syl5ibr 182	. . . 4 ⊢ ({⟨y, x⟩∣xRy} ⊆ R → (xRy → yRx))
8	2, 7	sylbi 174	. . 3 ⊢ (◡R ⊆ R → (xRy → yRx))
9	8	19.21aivv 944	. 2 ⊢ (◡R ⊆ R → ∀x∀y(xRy → yRx))
10		ssopab2 2119	. . . . 5 ⊢ ({⟨y, x⟩∣xRy} ⊆ {⟨y, x⟩∣yRx} ↔ ∀y∀x(xRy → yRx))
11		alcom 715	. . . . 5 ⊢ (∀y∀x(xRy → yRx) ↔ ∀x∀y(xRy → yRx))
12	10, 11	bitr 151	. . . 4 ⊢ ({⟨y, x⟩∣xRy} ⊆ {⟨y, x⟩∣yRx} ↔ ∀x∀y(xRy → yRx))
13		opabss 2100	. . . . 5 ⊢ {⟨y, x⟩∣yRx} ⊆ R
14		sstr2 1510	. . . . 5 ⊢ ({⟨y, x⟩∣xRy} ⊆ {⟨y, x⟩∣yRx} → ({⟨y, x⟩∣yRx} ⊆ R → {⟨y, x⟩∣xRy} ⊆ R))
15	13, 14	mpi 44	. . . 4 ⊢ ({⟨y, x⟩∣xRy} ⊆ {⟨y, x⟩∣yRx} → {⟨y, x⟩∣xRy} ⊆ R)
16	12, 15	sylbir 176	. . 3 ⊢ (∀x∀y(xRy → yRx) → {⟨y, x⟩∣xRy} ⊆ R)
17	16, 1	syl5ss 1544	. 2 ⊢ (∀x∀y(xRy → yRx) → ◡R ⊆ R)
18	9, 17	impbi 139	1 ⊢ (◡R ⊆ R ↔ ∀x∀y(xRy → yRx))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  {copab 2055  ^◡ccnv 2409

This theorem is referenced by:  er2 3201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426

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