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Theorem refsymrel2 34862
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 34814, cf. the comment of dfrefrels2 34812. (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 34814 . . . 4 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2 dfsymrel2 34844 . . . 4 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
31, 2anbi12i 622 . . 3 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
4 anandi3r 1134 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅𝑅𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
5 3anan32 1124 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅𝑅𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
63, 4, 53bitr2i 291 . 2 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
7 symrefref2 34858 . . . 4 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
87pm5.32ri 573 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅))
98anbi1i 619 . 2 (((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
106, 9bitri 267 1 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  w3a 1113  cin 3798  wss 3799   I cid 5250   × cxp 5341  ccnv 5342  dom cdm 5343  ran crn 5344  cres 5345  Rel wrel 5348   RefRel wrefrel 34531   SymRel wsymrel 34537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-refrel 34811  df-symrel 34839
This theorem is referenced by:  dfeqvrel2  34883  refrelred4  34925
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