dom_elim

Theorem dom_elim

Theorem. dom_elim
x ∈ dom R → ∃y. (x, y) ∈ R

Proof
01. 1 ⊢ x ∈ dom R, hypo.
02. 1 ⊢ x ∈ {x | ∃y. (x, y) ∈ R}, eq_subst dom_eq 1, P u ↔ x ∈ u.
03. 1 ⊢ ∃y. (x, y) ∈ R, comp_elim 2.
dom_elim. ⊢ x ∈ dom R → ∃y. (x, y) ∈ R, subj_intro 3.

Dependencies
The given proof depends on two axioms:
comp, eq_subst.