Definition df-trs 36613

Description: Define the class of all transitive sets (versus the transitive class defined in df-tr 5188). It is used only by df-trrels 36614.

Note the similarity of the definitions of df-refs 36555, df-syms 36583 and df-trs 36613. (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 36273	. 2 class Trs
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1538	. . . . . 6 class 𝑥
4	3	cdm 5580	. . . . . . 7 class dom 𝑥
5	3	crn 5581	. . . . . . 7 class ran 𝑥
6	4, 5	cxp 5578	. . . . . 6 class (dom 𝑥 × ran 𝑥)
7	3, 6	cin 3882	. . . . 5 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
8	7, 7	ccom 5584	. . . 4 class ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥)))
9		cssr 36263	. . . 4 class S
10	8, 7, 9	wbr 5070	. . 3 wff ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
11	10, 2	cab 2715	. 2 class {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
12	1, 11	wceq 1539	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  36616