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Theorem relcnveq 36012
 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 36011 . . 3 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
2 cnvsym 5947 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
31, 2bitr4di 293 . 2 (Rel 𝑅 → (𝑅 = 𝑅𝑅𝑅))
4 eqcom 2766 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
53, 4bitr3di 289 1 (Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1537   = wceq 1539   ⊆ wss 3859   class class class wbr 5033  ◡ccnv 5524  Rel wrel 5530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533 This theorem is referenced by:  relcnveq4  36014  cnvcosseq  36115  dfsymrel4  36220
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