dfrefrel2

Description: Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021)

		Ref	Expression
	Assertion	dfrefrel2	$⊢ RefRel R \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \land Rel R)$

Step	Hyp	Ref	Expression
1		df-refrel	$⊢ RefRel R \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \cap (dom R \times ran R) \land Rel R)$
2		dfrel6	$⊢ Rel R \leftrightarrow R \cap (dom R \times ran R) = R$
3	2	biimpi	$⊢ Rel R \to R \cap (dom R \times ran R) = R$
4	3	sseq2d	$⊢ Rel R \to (I \cap (dom R \times ran R) \subseteq R \cap (dom R \times ran R) \leftrightarrow I \cap (dom R \times ran R) \subseteq R)$
5	4	pm5.32ri	$⊢ (I \cap (dom R \times ran R) \subseteq R \cap (dom R \times ran R) \land Rel R) \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \land Rel R)$
6	1 5	bitri	$⊢ RefRel R \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \land Rel R)$

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