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Theorem dfsymrel4 34934
Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
dfsymrel4 ( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))

Proof of Theorem dfsymrel4
StepHypRef Expression
1 dfsymrel2 34932 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
2 relcnveq 34730 . . 3 (Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))
32pm5.32ri 571 . 2 ((𝑅𝑅 ∧ Rel 𝑅) ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))
41, 3bitri 267 1 ( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wss 3792  ccnv 5356  Rel wrel 5362   SymRel wsymrel 34627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369  df-symrel 34927
This theorem is referenced by:  symrelim  34942  idsymrel  34944  epnsymrel  34945
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