relcnveq2 - Mathbox for Peter Mazsa

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Theorem relcnveq2 38311

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)

**Assertion**
Ref	Expression
relcnveq2	⊢ (Rel 𝑅 → (^◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

Distinct variable group: 𝑥,𝑅,𝑦

**Proof of Theorem relcnveq2**
Step	Hyp	Ref	Expression
1		cnvsym 6085	. . . 4 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (Rel 𝑅 → (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		dfrel2 6162	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
4	3	biimpi 216	. . . . . 6 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
5	4	sseq1d 3978	. . . . 5 ⊢ (Rel 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
6		cnvsym 6085	. . . . 5 ⊢ (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥))
7	5, 6	bitr3di 286	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ⊆ ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥)))
8		relbrcnvg 6076	. . . . . 6 ⊢ (Rel 𝑅 → (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
9		relbrcnvg 6076	. . . . . 6 ⊢ (Rel 𝑅 → (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	8, 9	imbi12d 344	. . . . 5 ⊢ (Rel 𝑅 → ((𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
11	10	2albidv 1923	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
12	7, 11	bitrd 279	. . 3 ⊢ (Rel 𝑅 → (𝑅 ⊆ ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
13	2, 12	anbi12d 632	. 2 ⊢ (Rel 𝑅 → ((^◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ^◡𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))))
14		eqss 3962	. 2 ⊢ (^◡𝑅 = 𝑅 ↔ (^◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ^◡𝑅))
15		2albiim 1890	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
16	13, 14, 15	3bitr4g 314	1 ⊢ (Rel 𝑅 → (^◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ⊆ wss 3914 class class class wbr 5107 ^◡ccnv 5637 Rel wrel 5643

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 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

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 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-br 5108 df-opab 5170 df-xp 5644 df-rel 5645 df-cnv 5646

This theorem is referenced by: relcnveq4 38312

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