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Theorem relcnveq3 36383

Description: Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)

**Assertion**
Ref	Expression
relcnveq3	⊢ (Rel 𝑅 → (𝑅 = ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Distinct variable group: 𝑥,𝑅,𝑦

**Proof of Theorem relcnveq3**
Step	Hyp	Ref	Expression
1		eqss 3932	. 2 ⊢ (𝑅 = ^◡𝑅 ↔ (𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅))
2		cnvsym 6008	. . . . . . 7 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	2	biimpi 215	. . . . . 6 ⊢ (^◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	3	a1d 25	. . . . 5 ⊢ (^◡𝑅 ⊆ 𝑅 → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	adantl 481	. . . 4 ⊢ ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
6	5	com12 32	. . 3 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
7		dfrel2 6081	. . . . 5 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
8		cnvss 5770	. . . . . . . 8 ⊢ (^◡𝑅 ⊆ 𝑅 → ^◡^◡𝑅 ⊆ ^◡𝑅)
9		sseq1 3942	. . . . . . . 8 ⊢ (^◡^◡𝑅 = 𝑅 → (^◡^◡𝑅 ⊆ ^◡𝑅 ↔ 𝑅 ⊆ ^◡𝑅))
10	8, 9	syl5ibcom 244	. . . . . . 7 ⊢ (^◡𝑅 ⊆ 𝑅 → (^◡^◡𝑅 = 𝑅 → 𝑅 ⊆ ^◡𝑅))
11	2, 10	sylbir 234	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (^◡^◡𝑅 = 𝑅 → 𝑅 ⊆ ^◡𝑅))
12	11	com12 32	. . . . 5 ⊢ (^◡^◡𝑅 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ^◡𝑅))
13	7, 12	sylbi 216	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ^◡𝑅))
14	2	biimpri 227	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ^◡𝑅 ⊆ 𝑅)
15	13, 14	jca2 513	. . 3 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅)))
16	6, 15	impbid 211	. 2 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ^◡𝑅 ∧ ^◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
17	1, 16	syl5bb 282	1 ⊢ (Rel 𝑅 → (𝑅 = ^◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1537 = wceq 1539 ⊆ wss 3883 class class class wbr 5070 ^◡ccnv 5579 Rel wrel 5585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5586 df-rel 5587 df-cnv 5588

This theorem is referenced by: relcnveq 36384

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