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Theorem refrelredund4 36023
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35908) 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 5540 . . . . 5 ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅)
2 sstr2 3925 . . . . 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 617 . . 3 ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
5 dfrefrel2 35908 . . 3 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
64, 5sylibr 237 . 2 ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → RefRel 𝑅)
7 an12 644 . . 3 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
8 anandir 676 . . . . 5 (((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
9 refsymrel2 35956 . . . . 5 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
10 dfsymrel2 35938 . . . . . 6 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
1110anbi2i 625 . . . . 5 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (𝑅𝑅 ∧ Rel 𝑅)))
128, 9, 113bitr4i 306 . . . 4 (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅))
1312anbi2i 625 . . 3 (( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
147, 13bitr4i 281 . 2 (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))
15 df-redundp 36013 . 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 710 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 209  wa 399  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   redund wredundp 35628
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  df-redundp 36013
This theorem is referenced by:  refrelredund2  36024
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