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Theorem dfrefrel3 35636
Description: Alternate definition of the reflexive relation predicate. A relation is reflexive iff: for all elements on its domain and range, if an element of its domain is the same as an element of its range, then there is the relation between them.

Note that this is definitely not the definition we are accustomed to, like e.g. idref 6900 / idrefALT 5966 or df-reflexive 44795 (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, see refsymrels3 35682. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 35704, can we write the traditional form 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 for reflexive relations. For the special case with square Cartesian product when the two forms are equivalent see idinxpssinxp4 35458 where (∀𝑥𝐴𝑦𝐴(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 35649. (Contributed by Peter Mazsa, 8-Jul-2019.)

Assertion
Ref Expression
dfrefrel3 ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfrefrel3
StepHypRef Expression
1 dfrefrel2 35635 . 2 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2 idinxpss 35451 . . 3 (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
32anbi1i 623 . 2 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))
41, 3bitri 276 1 ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wral 3135  cin 3932  wss 3933   class class class wbr 5057   I cid 5452   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553   RefRel wrefrel 35340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-refrel 35632
This theorem is referenced by:  refsymrel3  35684
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