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Theorem refsymrels3 35981
 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 36003) can use the ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) version of dfrefrels3 35933, cf. the comment of dfrefrel3 35935. (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 35980 . 2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
2 idrefALT 5941 . . 3 (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3 cnvsym 5942 . . 3 (𝑟𝑟 ↔ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))
42, 3anbi12i 629 . 2 ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)))
51, 4rabbieq 35691 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∀wral 3106  {crab 3110   ∩ cin 3880   ⊆ wss 3881   class class class wbr 5031   I cid 5425  ◡ccnv 5519  dom cdm 5520   ↾ cres 5522   Rels crels 35634   RefRels crefrels 35637   SymRels csymrels 35643 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-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-rels 35904  df-ssr 35917  df-refs 35929  df-refrels 35930  df-syms 35957  df-symrels 35958 This theorem is referenced by: (None)
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