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Theorem cnvsym 6060

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 2160. (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 6052	. . 3 ⊢ Rel ^◡𝑅
2		ssrel3 5725	. . 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 5092	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑧^◡𝑅𝑥))
5		breq1 5092	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	4, 5	imbi12d 344	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧^◡𝑅𝑥 → 𝑧𝑅𝑥)))
7		breq2 5093	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑦^◡𝑅𝑧))
8		breq2 5093	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	7, 8	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦^◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2046	. 2 ⊢ (∀𝑦∀𝑥(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3440	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5821	. . . 4 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 349	. . 3 ⊢ ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1821	. 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 1539 ⊆ wss 3897 class class class wbr 5089 ^◡ccnv 5613 Rel wrel 5619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2710 df-cleq 2723 df-clel 2806 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-br 5090 df-opab 5152 df-xp 5620 df-rel 5621 df-cnv 5622

This theorem is referenced by: dfer2 8623 relcnveq3 38358 relcnveq 38359 relcnveq2 38360 cnvcosseq 38538 symrelcoss2 38567 dfsymrels3 38637 elrelscnveq3 38638 elrelscnveq 38639 elrelscnveq2 38640 dfsymrel3 38645 symrefref3 38659 refsymrels3 38661 elrefsymrels3 38665 dfeqvrels3 38684

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