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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 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) 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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