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Theorem cnvsym 6101

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 2157. (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 6091	. . 3 ⊢ Rel ^◡𝑅
2		ssrel3 5765	. . 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 5122	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑧^◡𝑅𝑥))
5		breq1 5122	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	4, 5	imbi12d 344	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧^◡𝑅𝑥 → 𝑧𝑅𝑥)))
7		breq2 5123	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑦^◡𝑅𝑧))
8		breq2 5123	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	7, 8	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦^◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2044	. 2 ⊢ (∀𝑦∀𝑥(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3463	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3463	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5862	. . . 4 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 349	. . 3 ⊢ ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
16	3, 10, 15	3bitri 297	1 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ⊆ wss 3926 class class class wbr 5119 ^◡ccnv 5653 Rel wrel 5659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-br 5120 df-opab 5182 df-xp 5660 df-rel 5661 df-cnv 5662

This theorem is referenced by: dfer2 8720 relcnveq3 38339 relcnveq 38340 relcnveq2 38341 cnvcosseq 38455 symrelcoss2 38484 elrelscnveq3 38509 elrelscnveq 38510 elrelscnveq2 38511 dfsymrels3 38564 dfsymrel3 38568 symrefref3 38582 refsymrels3 38584 elrefsymrels3 38588 dfeqvrels3 38607

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