relcnveq2 - Mathbox for Peter Mazsa

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Theorem relcnveq2 38711

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 6071	. . . 4 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (Rel 𝑅 → (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		dfrel2 6144	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
4	3	biimpi 218	. . . . . 6 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
5	4	sseq1d 3948	. . . . 5 ⊢ (Rel 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
6		cnvsym 6071	. . . . 5 ⊢ (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥))
7	5, 6	bitr3di 288	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ⊆ ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥)))
8		relbrcnvg 6064	. . . . . 6 ⊢ (Rel 𝑅 → (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
9		relbrcnvg 6064	. . . . . 6 ⊢ (Rel 𝑅 → (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	8, 9	imbi12d 346	. . . . 5 ⊢ (Rel 𝑅 → ((𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
11	10	2albidv 1931	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
12	7, 11	bitrd 281	. . 3 ⊢ (Rel 𝑅 → (𝑅 ⊆ ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
13	2, 12	anbi12d 639	. 2 ⊢ (Rel 𝑅 → ((^◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ^◡𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))))
14		eqss 3932	. 2 ⊢ (^◡𝑅 = 𝑅 ↔ (^◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ^◡𝑅))
15		2albiim 1898	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
16	13, 14, 15	3bitr4g 316	1 ⊢ (Rel 𝑅 → (^◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 ∀wal 1546 = wceq 1548 ⊆ wss 3885 class class class wbr 5075 ^◡ccnv 5620 Rel wrel 5626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-ext 2713 ax-sep 5221 ax-pr 5365

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-sb 2075 df-clab 2720 df-cleq 2733 df-clel 2816 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-br 5076 df-opab 5138 df-xp 5627 df-rel 5628 df-cnv 5629

This theorem is referenced by: relcnveq4 38712

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