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Theorem relcnveq 36994
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.)
Assertion
Ref Expression
relcnveq (Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))

Proof of Theorem relcnveq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnveq3 36993 . . 3 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
2 cnvsym 6102 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
31, 2bitr4di 288 . 2 (Rel 𝑅 → (𝑅 = 𝑅𝑅𝑅))
4 eqcom 2738 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
53, 4bitr3di 285 1 (Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wss 3944   class class class wbr 5141  ccnv 5668  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677
This theorem is referenced by:  relcnveq4  36996  cnvcosseq  37110  dfsymrel4  37224
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