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Theorem refrelredund4 35885
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35770) 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 5572 . . . . 5 ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅)
2 sstr2 3974 . . . . 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 616 . . 3 ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
5 dfrefrel2 35770 . . 3 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
64, 5sylibr 236 . 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 35818 . . . . 5 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
10 dfsymrel2 35800 . . . . . 6 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
1110anbi2i 624 . . . . 5 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
128, 9, 113bitr4i 305 . . . 4 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅))
1312anbi2i 624 . . 3 (( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
147, 13bitr4i 280 . 2 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))
15 df-redundp 35875 . 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 208  wa 398  cin 3935  wss 3936   I cid 5459   × cxp 5553  ccnv 5554  dom cdm 5555  ran crn 5556  cres 5557  Rel wrel 5560   RefRel wrefrel 35474   SymRel wsymrel 35480   redund wredundp 35490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-refrel 35767  df-symrel 35795  df-redundp 35875
This theorem is referenced by:  refrelredund2  35886
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