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Metamath Proof Explorer

Theorem dmhashres

Description: Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017)

		Ref	Expression
	Assertion	dmhashres	$⊢ dom ({\|.\| ↾}_{A}) = A$

Step	Hyp	Ref	Expression
1		dmres	$⊢ dom ({\|.\| ↾}_{A}) = A \cap dom \|.\|$
2		hashf	$⊢ \|.\| : V ⟶ ℕ_{0} \cup \{+∞\}$
3	2	fdmi	$⊢ dom \|.\| = V$
4	3	ineq2i	$⊢ A \cap dom \|.\| = A \cap V$
5		inv1	$⊢ A \cap V = A$
6	1 4 5	3eqtri	$⊢ dom ({\|.\| ↾}_{A}) = A$