Definition df-trs 39407

Description: Define the class of all transitive sets (versus the transitive class defined in df-tr 5213). It is used only by df-trrels 39408.

Note the similarity of the definitions of df-refs 39341, df-syms 39373 and df-trs 39407. (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

Step	Hyp	Ref	Expression
1		ctrs 38947	. 2 class Trs
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1569	. . . . . 6 class 𝑥
4	3	cdm 5655	. . . . . . 7 class dom 𝑥
5	3	crn 5656	. . . . . . 7 class ran 𝑥
6	4, 5	cxp 5653	. . . . . 6 class (dom 𝑥 × ran 𝑥)
7	3, 6	cin 3898	. . . . 5 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
8	7, 7	ccom 5659	. . . 4 class ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥)))
9		cssr 38937	. . . 4 class S
10	8, 7, 9	wbr 5103	. . 3 wff ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
11	10, 2	cab 2738	. 2 class {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
12	1, 11	wceq 1570	1 wff Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

This definition is used by:  dftrrels2  39410