Theorem elrelscnveq 39004

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)

Assertion

Ref	Expression
elrelscnveq	⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Proof of Theorem elrelscnveq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrelscnveq3 39003	. . 3 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
2		cnvsym 6065	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	1, 2	bitr4di 290	. 2 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅))
4		eqcom 2746	. 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅)
5	3, 4	bitr3di 287	1 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883   class class class wbr 5073  ^◡ccnv 5618   Rels crels 38561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-xp 5625  df-rel 5626  df-cnv 5627  df-rels 38816

This theorem is referenced by:  elrelscnveq4  39006  dfsymrels4  39007