Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 6910 / idrefALT 5975 or df-reflexive 44874 ⊢ (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 35804. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 35826, 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 35579 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 35771. (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 35757 | . 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)) | |
2 | idinxpss 35572 | . . 3 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦)) | |
3 | 2 | anbi1i 625 | . 2 ⊢ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅)) |
4 | 1, 3 | bitri 277 | 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 208 ∧ wa 398 ∀wral 3140 ∩ cin 3937 ⊆ wss 3938 class class class wbr 5068 I cid 5461 × cxp 5555 dom cdm 5557 ran crn 5558 Rel wrel 5562 RefRel wrefrel 35461 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-refrel 35754 |
This theorem is referenced by: refsymrel3 35806 |
Copyright terms: Public domain | W3C validator |