relcnveq2 - Mathbox for Peter Mazsa

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Theorem relcnveq2 37192

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 6114	. . . 4 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (Rel 𝑅 → (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		dfrel2 6189	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
4	3	biimpi 215	. . . . . 6 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
5	4	sseq1d 4014	. . . . 5 ⊢ (Rel 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
6		cnvsym 6114	. . . . 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 6105	. . . . . 6 ⊢ (Rel 𝑅 → (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
9		relbrcnvg 6105	. . . . . 6 ⊢ (Rel 𝑅 → (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	8, 9	imbi12d 345	. . . . 5 ⊢ (Rel 𝑅 → ((𝑥^◡𝑅𝑦 → 𝑦^◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
11	10	2albidv 1927	. . . 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 3998	. 2 ⊢ (^◡𝑅 = 𝑅 ↔ (^◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ^◡𝑅))
15		2albiim 1894	. 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 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ⊆ wss 3949 class class class wbr 5149 ^◡ccnv 5676 Rel wrel 5682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685

This theorem is referenced by: relcnveq4 37193

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