Theorem relcnveq3 38709

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 6071	. . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	2	biimpi 218	. . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	3	a1d 25	. . . . 5 ⊢ (◡𝑅 ⊆ 𝑅 → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	adantl 483	. . . 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 6144	. . . . 5 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8		cnvss 5817	. . . . . . . 8 ⊢ (◡𝑅 ⊆ 𝑅 → ◡◡𝑅 ⊆ ◡𝑅)
9		sseq1 3942	. . . . . . . 8 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅))
10	8, 9	syl5ibcom 247	. . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅))
11	2, 10	sylbir 237	. . . . . 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 219	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅))
14	2	biimpri 230	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ◡𝑅 ⊆ 𝑅)
15	13, 14	jca2 519	. . 3 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅)))
16	6, 15	impbid 214	. 2 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
17	1, 16	bitrid 285	1 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

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-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:  relcnveq  38710