Theorem relcnveq 35594

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		eqcom 2828	. 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅)
2		relcnveq3 35593	. . 3 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		cnvsym 5974	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	2, 3	syl6bbr 291	. 2 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅))
5	1, 4	syl5rbbr 288	1 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537   ⊆ wss 3936   class class class wbr 5066  ^◡ccnv 5554  Rel wrel 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563

This theorem is referenced by:  relcnveq4  35596  cnvcosseq  35697  dfsymrel4  35802