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Theorem refrelredund4 37310
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 37190) 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 5686 . . . . 5 ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅)
2 sstr2 3985 . . . . 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 37190 . . 3 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
64, 5sylibr 233 . 2 ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → RefRel 𝑅)
7 an12 643 . . 3 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
8 anandir 675 . . . . 5 (((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
9 refsymrel2 37242 . . . . 5 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
10 dfsymrel2 37224 . . . . . 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 302 . . . 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 277 . 2 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))
15 df-redundp 37300 . 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 709 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 205  wa 396  cin 3943  wss 3944   I cid 5566   × cxp 5667  ccnv 5668  dom cdm 5669  ran crn 5670  cres 5671  Rel wrel 5674   RefRel wrefrel 36854   SymRel wsymrel 36860   redund wredundp 36870
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-refrel 37187  df-symrel 37219  df-redundp 37300
This theorem is referenced by:  refrelredund2  37311
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