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Metamath Proof Explorer

Theorem harval

Description: Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015)

		Ref	Expression
	Assertion	harval	$⊢ X \in V \to har (X) = \{y \in On \| y ≼ X\}$

Step	Hyp	Ref	Expression
1		elex	$⊢ X \in V \to X \in V$
2		breq2	$⊢ x = X \to (y ≼ x \leftrightarrow y ≼ X)$
3	2	rabbidv	$⊢ x = X \to \{y \in On \| y ≼ x\} = \{y \in On \| y ≼ X\}$
4		df-har	$⊢ har = (x \in V ⟼ \{y \in On \| y ≼ x\})$
5		hartogs	$⊢ x \in V \to \{y \in On \| y ≼ x\} \in On$
6	3 4 5	fvmpt3	$⊢ X \in V \to har (X) = \{y \in On \| y ≼ X\}$
7	1 6	syl	$⊢ X \in V \to har (X) = \{y \in On \| y ≼ X\}$