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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-symrel | Structured version Visualization version GIF version |
Description: Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 37051) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 37065. Alternate definitions are dfsymrel2 37057 and dfsymrel3 37058. (Contributed by Peter Mazsa, 16-Jul-2021.) |
Ref | Expression |
---|---|
df-symrel | ⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cR | . . 3 class 𝑅 | |
2 | 1 | wsymrel 36692 | . 2 wff SymRel 𝑅 |
3 | 1 | cdm 5634 | . . . . . . 7 class dom 𝑅 |
4 | 1 | crn 5635 | . . . . . . 7 class ran 𝑅 |
5 | 3, 4 | cxp 5632 | . . . . . 6 class (dom 𝑅 × ran 𝑅) |
6 | 1, 5 | cin 3910 | . . . . 5 class (𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
7 | 6 | ccnv 5633 | . . . 4 class ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
8 | 7, 6 | wss 3911 | . . 3 wff ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) |
9 | 1 | wrel 5639 | . . 3 wff Rel 𝑅 |
10 | 8, 9 | wa 397 | . 2 wff (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) |
11 | 2, 10 | wb 205 | 1 wff ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsymrel2 37057 |
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