Definition df-trs 37437

Description: Define the class of all transitive sets (versus the transitive class defined in df-tr 5266). It is used only by df-trrels 37438.

Note the similarity of the definitions of df-refs 37375, df-syms 37407 and df-trs 37437. (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 37051	. 2 class Trs
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1540	. . . . . 6 class 𝑥
4	3	cdm 5676	. . . . . . 7 class dom 𝑥
5	3	crn 5677	. . . . . . 7 class ran 𝑥
6	4, 5	cxp 5674	. . . . . 6 class (dom 𝑥 × ran 𝑥)
7	3, 6	cin 3947	. . . . 5 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
8	7, 7	ccom 5680	. . . 4 class ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥)))
9		cssr 37041	. . . 4 class S
10	8, 7, 9	wbr 5148	. . 3 wff ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
11	10, 2	cab 2709	. 2 class {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
12	1, 11	wceq 1541	1 wff Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

This definition is referenced by:  dftrrels2  37440