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Theorem refsymrel2 38095
Description: A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrel2 38043, cf. the comment of dfrefrels2 38041. (Contributed by Peter Mazsa, 23-Aug-2021.)
Assertion
Ref Expression
refsymrel2 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))

Proof of Theorem refsymrel2
StepHypRef Expression
1 dfrefrel2 38043 . . . 4 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2 dfsymrel2 38077 . . . 4 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
31, 2anbi12i 626 . . 3 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
4 anandi3r 1100 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅𝑅𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
5 3anan32 1094 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅𝑅𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
63, 4, 53bitr2i 298 . 2 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
7 symrefref2 38091 . . . 4 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
87pm5.32ri 574 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅))
98anbi1i 622 . 2 (((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
106, 9bitri 274 1 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1084  cin 3938  wss 3939   I cid 5569   × cxp 5670  ccnv 5671  dom cdm 5672  ran crn 5673  cres 5674  Rel wrel 5677   RefRel wrefrel 37711   SymRel wsymrel 37717
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-ext 2696  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 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 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  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-refrel 38040  df-symrel 38072
This theorem is referenced by:  dfeqvrel2  38118  refrelredund4  38163
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