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Theorem cnvsym 5941

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 2160	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2		relcnv 5934	. . 3 ⊢ Rel ^◡𝑅
3		ssrel 5621	. . 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 3444	. . . . . 6 ⊢ 𝑦 ∈ V
6		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5717	. . . . 5 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8		df-br 5031	. . . . 5 ⊢ (𝑦^◡𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅)
9	7, 8	bitr3i 280	. . . 4 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅)
10		df-br 5031	. . . 4 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
11	9, 10	imbi12i 354	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12	11	2albii 1822	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
13	1, 4, 12	3bitr4i 306	1 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 ⊆ wss 3881 ⟨cop 4531 class class class wbr 5030 ^◡ccnv 5518 Rel wrel 5524

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527

This theorem is referenced by: dfer2 8273 relcnveq3 35738 relcnveq 35739 relcnveq2 35740 cnvcosseq 35842 symrelcoss2 35866 elrelscnveq3 35891 elrelscnveq 35892 elrelscnveq2 35893 dfsymrels3 35942 dfsymrel3 35946 symrefref3 35960 refsymrels3 35962 elrefsymrels3 35966 dfeqvrels3 35984

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