Theorem elrelscnveq 38490

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 38489	. . 3 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
2		cnvsym 6088	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	1, 2	bitr4di 289	. 2 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅))
4		eqcom 2737	. 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅)
5	3, 4	bitr3di 286	1 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917   class class class wbr 5110  ^◡ccnv 5640   Rels crels 38178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-rels 38483

This theorem is referenced by:  elrelscnveq4  38492  dfsymrels4  38545