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Theorem refrelredund4 38637
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38517) if the relation is symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)
Assertion
Ref Expression
refrelredund4 redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))

Proof of Theorem refrelredund4
StepHypRef Expression
1 inxpssres 5701 . . . . 5 ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅)
2 sstr2 3989 . . . . 5 (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅) → (( I ↾ dom 𝑅) ⊆ 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
31, 2ax-mp 5 . . . 4 (( I ↾ dom 𝑅) ⊆ 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅)
43anim1i 615 . . 3 ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
5 dfrefrel2 38517 . . 3 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
64, 5sylibr 234 . 2 ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → RefRel 𝑅)
7 an12 645 . . 3 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
8 anandir 677 . . . . 5 (((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
9 refsymrel2 38569 . . . . 5 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
10 dfsymrel2 38551 . . . . . 6 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
1110anbi2i 623 . . . . 5 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
128, 9, 113bitr4i 303 . . . 4 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅))
1312anbi2i 623 . . 3 (( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
147, 13bitr4i 278 . 2 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))
15 df-redundp 38627 . 2 ( redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → RefRel 𝑅) ∧ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))))
166, 14, 15mpbir2an 711 1 redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  cin 3949  wss 3950   I cid 5576   × cxp 5682  ccnv 5683  dom cdm 5684  ran crn 5685  cres 5686  Rel wrel 5689   RefRel wrefrel 38189   SymRel wsymrel 38195   redund wredundp 38205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-refrel 38514  df-symrel 38546  df-redundp 38627
This theorem is referenced by:  refrelredund2  38638
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