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Definition df-trs 35694
Description: Define the class of all transitive sets (versus the transitive class defined in df-tr 5170). It is used only by df-trrels 35695.

Note the similarity of the definitions of df-refs 35636, df-syms 35664 and df-trs 35694. (Contributed by Peter Mazsa, 17-Jul-2021.)

Assertion
Ref Expression
df-trs Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Detailed syntax breakdown of Definition df-trs
StepHypRef Expression
1 ctrs 35353 . 2 class Trs
2 vx . . . . . . 7 setvar 𝑥
32cv 1529 . . . . . 6 class 𝑥
43cdm 5554 . . . . . . 7 class dom 𝑥
53crn 5555 . . . . . . 7 class ran 𝑥
64, 5cxp 5552 . . . . . 6 class (dom 𝑥 × ran 𝑥)
73, 6cin 3939 . . . . 5 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
87, 7ccom 5558 . . . 4 class ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥)))
9 cssr 35343 . . . 4 class S
108, 7, 9wbr 5063 . . 3 wff ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
1110, 2cab 2804 . 2 class {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
121, 11wceq 1530 1 wff Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Colors of variables: wff setvar class
This definition is referenced by:  dftrrels2  35697
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