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Theorem refsymrel2 35956
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 35908, cf. the comment of dfrefrels2 35906. (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 35908 . . . 4 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2 dfsymrel2 35938 . . . 4 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
31, 2anbi12i 629 . . 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 302 . 2 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
7 symrefref2 35952 . . . 4 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
87pm5.32ri 579 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅))
98anbi1i 626 . 2 (((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
106, 9bitri 278 1 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084  cin 3883  wss 3884   I cid 5427   × cxp 5521  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  Rel wrel 5528   RefRel wrefrel 35612   SymRel wsymrel 35618
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-refrel 35905  df-symrel 35933
This theorem is referenced by:  dfeqvrel2  35978  refrelredund4  36023
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