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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 7166 / idrefALT 6134 or df-reflexive 48999 ⊢ (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 38548. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 38571, 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 38302 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 38513. (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 38497 | . 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)) | |
2 | idinxpss 38294 | . . 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 275 | 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 206 ∧ wa 395 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 I cid 5582 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 RefRel wrefrel 38168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-refrel 38494 |
This theorem is referenced by: refsymrel3 38550 |
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