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Theorem dfsymrel2 35964
 Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
dfsymrel2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))

Proof of Theorem dfsymrel2
StepHypRef Expression
1 df-symrel 35959 . 2 ( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 35783 . . . . . 6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 219 . . . . 5 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43cnveqd 5711 . . . 4 (Rel 𝑅(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
54, 3sseq12d 3948 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑅))
65pm5.32ri 579 . 2 (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (𝑅𝑅 ∧ Rel 𝑅))
71, 6bitri 278 1 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∩ cin 3880   ⊆ wss 3881   × cxp 5518  ◡ccnv 5519  dom cdm 5520  ran crn 5521  Rel wrel 5525   SymRel wsymrel 35644 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-symrel 35959 This theorem is referenced by:  dfsymrel3  35965  dfsymrel4  35966  dfsymrel5  35967  elsymrelsrel  35972  symreleq  35973  symrelcoss  35975  refsymrel2  35982  eqvrelsym  36019  refrelredund4  36049
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