elrelscnveq3 - Mathbox for Peter Mazsa

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Theorem elrelscnveq3 38447

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 4024	. 2 ⊢ (𝑅 = ^◡𝑅 ↔ (𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅))
2		cnvsym 6144	. . . . . . 7 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	2	biimpi 216	. . . . . 6 ⊢ (^◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	3	a1d 25	. . . . 5 ⊢ (^◡𝑅 ⊆ 𝑅 → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	adantl 481	. . . 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 38444	. . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
8		dfrel2 6220	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝑅 ∈ Rels → ^◡^◡𝑅 = 𝑅)
10		cnvss 5897	. . . . . . 7 ⊢ (^◡𝑅 ⊆ 𝑅 → ^◡^◡𝑅 ⊆ ^◡𝑅)
11		sseq1 4034	. . . . . . 7 ⊢ (^◡^◡𝑅 = 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
12	10, 11	syl5ibcom 245	. . . . . 6 ⊢ (^◡𝑅 ⊆ 𝑅 → (^◡^◡𝑅 = 𝑅 → 𝑅 ⊆ ^◡𝑅))
13	2, 12	sylbir 235	. . . . 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 228	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ^◡𝑅 ⊆ 𝑅)
16	14, 15	jca2 513	. . 3 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅)))
17	6, 16	impbid 212	. 2 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
18	1, 17	bitrid 283	1 ⊢ (𝑅 ∈ Rels → (𝑅 = ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 class class class wbr 5166 ^◡ccnv 5699 Rel wrel 5705 Rels crels 38137

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-rels 38441

This theorem is referenced by: elrelscnveq 38448

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