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Theorem elrefsymrels3 38094
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 38113) can use the 𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrels3 38038, cf. the comment of dfrefrel3 38040. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
elrefsymrels3 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrefsymrels3
StepHypRef Expression
1 elrefsymrels2 38093 . 2 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
2 idrefALT 6113 . . . 4 (( I ↾ dom 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)
3 cnvsym 6114 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
42, 3anbi12i 626 . . 3 ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
54anbi1i 622 . 2 (((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
61, 5bitri 274 1 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wcel 2098  wral 3051  cin 3940  wss 3941   class class class wbr 5144   I cid 5570  ccnv 5672  dom cdm 5673  cres 5675   Rels crels 37703   RefRels crefrels 37706   SymRels csymrels 37712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-rels 38009  df-ssr 38022  df-refs 38034  df-refrels 38035  df-syms 38066  df-symrels 38067
This theorem is referenced by: (None)
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