Theorem relcnveq 38834

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 38833	. . 3 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
2		cnvsym 6104	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	1, 2	bitr4di 292	. 2 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅))
4		eqcom 2772	. 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅)
5	3, 4	bitr3di 289	1 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ⊆ wss 3907   class class class wbr 5104  ^◡ccnv 5650  Rel wrel 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659

This theorem is referenced by:  relcnveq4  38836  cnvcosseq  39033  dfsymrel4  39141