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Theorem refsymrel2 37058
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 37006, cf. the comment of dfrefrels2 37004. (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 37006 . . . 4 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2 dfsymrel2 37040 . . . 4 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
31, 2anbi12i 628 . . 3 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
4 anandi3r 1104 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅𝑅𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
5 3anan32 1098 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅𝑅𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
63, 4, 53bitr2i 299 . 2 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
7 symrefref2 37054 . . . 4 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
87pm5.32ri 577 . . 3 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅))
98anbi1i 625 . 2 (((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
106, 9bitri 275 1 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088  cin 3914  wss 3915   I cid 5535   × cxp 5636  ccnv 5637  dom cdm 5638  ran crn 5639  cres 5640  Rel wrel 5643   RefRel wrefrel 36669   SymRel wsymrel 36675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-refrel 37003  df-symrel 37035
This theorem is referenced by:  dfeqvrel2  37081  refrelredund4  37126
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