cnvsym - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cnvsym		Structured version Visualization version GIF version

Theorem cnvsym 6119

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.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2146. (Revised by BTernaryTau, 29-Dec-2024.)

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

Distinct variable group: 𝑥,𝑦,𝑅

Proof of Theorem cnvsym Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6109	. . 3 ⊢ Rel ^◡𝑅
2		ssrel3 5788	. . 3 ⊢ (Rel ^◡𝑅 → (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥)))
3	1, 2	ax-mp 5	. 2 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥))
4		breq1 5152	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑧^◡𝑅𝑥))
5		breq1 5152	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	4, 5	imbi12d 343	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧^◡𝑅𝑥 → 𝑧𝑅𝑥)))
7		breq2 5153	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑦^◡𝑅𝑧))
8		breq2 5153	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	7, 8	imbi12d 343	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦^◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2039	. 2 ⊢ (∀𝑦∀𝑥(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3465	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3465	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5885	. . . 4 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 348	. . 3 ⊢ ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1814	. 2 ⊢ (∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
16	3, 10, 15	3bitri 296	1 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ⊆ wss 3944 class class class wbr 5149 ^◡ccnv 5677 Rel wrel 5683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686

This theorem is referenced by: dfer2 8726 relcnveq3 37923 relcnveq 37924 relcnveq2 37925 cnvcosseq 38039 symrelcoss2 38068 elrelscnveq3 38093 elrelscnveq 38094 elrelscnveq2 38095 dfsymrels3 38148 dfsymrel3 38152 symrefref3 38166 refsymrels3 38168 elrefsymrels3 38172 dfeqvrels3 38191

W3C validator