elrelscnveq3 - Mathbox for Peter Mazsa

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Theorem elrelscnveq3 36725

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)

**Assertion**
Ref	Expression
elrelscnveq3	⊢ (𝑅 ∈ Rels → (𝑅 = ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Distinct variable group: 𝑥,𝑅,𝑦

**Proof of Theorem elrelscnveq3**
Step	Hyp	Ref	Expression
1		eqss 3946	. 2 ⊢ (𝑅 = ^◡𝑅 ↔ (𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅))
2		cnvsym 6040	. . . . . . 7 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	2	biimpi 215	. . . . . 6 ⊢ (^◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	3	a1d 25	. . . . 5 ⊢ (^◡𝑅 ⊆ 𝑅 → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	adantl 482	. . . 4 ⊢ ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
6	5	com12 32	. . 3 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
7		elrelsrelim 36722	. . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
8		dfrel2 6115	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
9	7, 8	sylib 217	. . . . 5 ⊢ (𝑅 ∈ Rels → ^◡^◡𝑅 = 𝑅)
10		cnvss 5802	. . . . . . 7 ⊢ (^◡𝑅 ⊆ 𝑅 → ^◡^◡𝑅 ⊆ ^◡𝑅)
11		sseq1 3956	. . . . . . 7 ⊢ (^◡^◡𝑅 = 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
12	10, 11	syl5ibcom 244	. . . . . 6 ⊢ (^◡𝑅 ⊆ 𝑅 → (^◡^◡𝑅 = 𝑅 → 𝑅 ⊆ ^◡𝑅))
13	2, 12	sylbir 234	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (^◡^◡𝑅 = 𝑅 → 𝑅 ⊆ ^◡𝑅))
14	9, 13	syl5com 31	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ^◡𝑅))
15	2	biimpri 227	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ^◡𝑅 ⊆ 𝑅)
16	14, 15	jca2 514	. . 3 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅)))
17	6, 16	impbid 211	. 2 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
18	1, 17	bitrid 282	1 ⊢ (𝑅 ∈ Rels → (𝑅 = ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1538 = wceq 1540 ∈ wcel 2105 ⊆ wss 3897 class class class wbr 5087 ^◡ccnv 5607 Rel wrel 5613 Rels crels 36407

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2708 ax-sep 5238 ax-nul 5245 ax-pr 5367

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3405 df-v 3443 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-br 5088 df-opab 5150 df-xp 5614 df-rel 5615 df-cnv 5616 df-rels 36719

This theorem is referenced by: elrelscnveq 36726

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