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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrefrel3 | Structured version Visualization version GIF version |
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 7091 / idrefALT 6065 or df-reflexive 47184 ⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, see refsymrels3 37018. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 37041, 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 36771 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 36983. (Contributed by Peter Mazsa, 8-Jul-2019.) |
Ref | Expression |
---|---|
dfrefrel3 | ⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrefrel2 36967 | . 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)) | |
2 | idinxpss 36763 | . . 3 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦)) | |
3 | 2 | anbi1i 624 | . 2 ⊢ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅)) |
4 | 1, 3 | bitri 274 | 1 ⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wral 3064 ∩ cin 3909 ⊆ wss 3910 class class class wbr 5105 I cid 5530 × cxp 5631 dom cdm 5633 ran crn 5634 Rel wrel 5638 RefRel wrefrel 36630 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-dm 5643 df-rn 5644 df-res 5645 df-refrel 36964 |
This theorem is referenced by: refsymrel3 37020 |
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