relcnveq2 - Mathbox for Peter Mazsa

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Theorem relcnveq2 38360

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 6060	. . . 4 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (Rel 𝑅 → (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		dfrel2 6136	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
4	3	biimpi 216	. . . . . 6 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
5	4	sseq1d 3961	. . . . 5 ⊢ (Rel 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
6		cnvsym 6060	. . . . 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 6053	. . . . . 6 ⊢ (Rel 𝑅 → (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
9		relbrcnvg 6053	. . . . . 6 ⊢ (Rel 𝑅 → (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	8, 9	imbi12d 344	. . . . 5 ⊢ (Rel 𝑅 → ((𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
11	10	2albidv 1924	. . . 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 3945	. 2 ⊢ (^◡𝑅 = 𝑅 ↔ (^◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ^◡𝑅))
15		2albiim 1891	. 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 1539 = wceq 1541 ⊆ 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-ral 3048 df-rex 3057 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: relcnveq4 38361

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