Theorem relcnveq 38323

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)

Assertion

Ref	Expression
relcnveq	⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Proof of Theorem relcnveq

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

Step	Hyp	Ref	Expression
1		relcnveq3 38322	. . 3 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
2		cnvsym 6132	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	1, 2	bitr4di 289	. 2 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅))
4		eqcom 2744	. 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅)
5	3, 4	bitr3di 286	1 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540   ⊆ wss 3951   class class class wbr 5143  ^◡ccnv 5684  Rel wrel 5690

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693

This theorem is referenced by:  relcnveq4  38325  cnvcosseq  38438  dfsymrel4  38552