Theorem relcnveq3 39004

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 3951	. 2 ⊢ (𝑅 = ◡𝑅 ↔ (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅))
2		cnvsym 6113	. . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	2	biimpi 219	. . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	3	a1d 26	. . . . 5 ⊢ (◡𝑅 ⊆ 𝑅 → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	adantl 486	. . . 4 ⊢ ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
6	5	com12 33	. . 3 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
7		dfrel2 6186	. . . . 5 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8		cnvss 5857	. . . . . . . 8 ⊢ (◡𝑅 ⊆ 𝑅 → ◡◡𝑅 ⊆ ◡𝑅)
9		sseq1 3961	. . . . . . . 8 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅))
10	8, 9	syl5ibcom 248	. . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅))
11	2, 10	sylbir 238	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅))
12	11	com12 33	. . . . 5 ⊢ (◡◡𝑅 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅))
13	7, 12	sylbi 220	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅))
14	2	biimpri 231	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ◡𝑅 ⊆ 𝑅)
15	13, 14	jca2 522	. . 3 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅)))
16	6, 15	impbid 215	. 2 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
17	1, 16	bitrid 286	1 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567   = wceq 1569   ⊆ wss 3904   class class class wbr 5108  ^◡ccnv 5659  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668

This theorem is used by:  relcnveq  39005