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Theorem refsymrels3 38274
Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 38297) can use the 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) version of dfrefrels3 38222, cf. the comment of dfrefrel3 38224. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
refsymrels3 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem refsymrels3
StepHypRef Expression
1 refsymrels2 38273 . 2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
2 idrefALT 6113 . . 3 (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3 cnvsym 6114 . . 3 (𝑟𝑟 ↔ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))
42, 3anbi12i 626 . 2 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)))
51, 4rabbieq 3428 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1532   = wceq 1534  wral 3051  {crab 3419  cin 3945  wss 3946   class class class wbr 5143   I cid 5569  ccnv 5671  dom cdm 5672  cres 5674   Rels crels 37888   RefRels crefrels 37891   SymRels csymrels 37897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-rels 38193  df-ssr 38206  df-refs 38218  df-refrels 38219  df-syms 38250  df-symrels 38251
This theorem is referenced by: (None)
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