Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > df-refrel | Structured version Visualization version GIF version |
Description: Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 36630. Alternate definitions are dfrefrel2 36629 and dfrefrel3 36630. For sets, being an element of the class of reflexive relations (df-refrels 36625) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 36633. (Contributed by Peter Mazsa, 16-Jul-2021.) |
Ref | Expression |
---|---|
df-refrel | ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cR | . . 3 class 𝑅 | |
2 | 1 | wrefrel 36335 | . 2 wff RefRel 𝑅 |
3 | cid 5489 | . . . . 5 class I | |
4 | 1 | cdm 5590 | . . . . . 6 class dom 𝑅 |
5 | 1 | crn 5591 | . . . . . 6 class ran 𝑅 |
6 | 4, 5 | cxp 5588 | . . . . 5 class (dom 𝑅 × ran 𝑅) |
7 | 3, 6 | cin 3891 | . . . 4 class ( I ∩ (dom 𝑅 × ran 𝑅)) |
8 | 1, 6 | cin 3891 | . . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
9 | 7, 8 | wss 3892 | . . 3 wff ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
10 | 1 | wrel 5595 | . . 3 wff Rel 𝑅 |
11 | 9, 10 | wa 396 | . 2 wff (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) |
12 | 2, 11 | wb 205 | 1 wff ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
This definition is referenced by: dfrefrel2 36629 refrelid 36635 |
Copyright terms: Public domain | W3C validator |