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Theorem refsymrel3 35913
 Description: A relation which is reflexive and symmetric (like an equivalence relation) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for its reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrel3 35865, cf. the comment of dfrefrel3 35865. (Contributed by Peter Mazsa, 23-Aug-2021.)
Assertion
Ref Expression
refsymrel3 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem refsymrel3
StepHypRef Expression
1 dfrefrel3 35865 . . . 4 ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))
2 dfsymrel3 35895 . . . 4 ( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
31, 2anbi12i 629 . . 3 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅) ∧ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅)))
4 anandi3r 1100 . . 3 ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅) ∧ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅)))
5 3anan32 1094 . . 3 ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
63, 4, 53bitr2i 302 . 2 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
7 symrefref3 35909 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
87pm5.32ri 579 . . 3 ((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
98anbi1i 626 . 2 (((∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
106, 9bitri 278 1 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536  ∀wral 3133   class class class wbr 5052  dom cdm 5542  ran crn 5543  Rel wrel 5547   RefRel wrefrel 35568   SymRel wsymrel 35574 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-refrel 35861  df-symrel 35889 This theorem is referenced by:  dfeqvrel3  35935
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