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Metamath Proof Explorer

Theorem inpreima

Description: Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 14-Jun-2016)

		Ref	Expression
	Assertion	inpreima	$⊢ Fun F \to F^{-1} [A \cap B] = F^{-1} [A] \cap F^{-1} [B]$

Step	Hyp	Ref	Expression
1		funcnvcnv	$⊢ Fun F \to Fun {F^{-1}}^{-1}$
2		imain	$⊢ Fun {F^{-1}}^{-1} \to F^{-1} [A \cap B] = F^{-1} [A] \cap F^{-1} [B]$
3	1 2	syl	$⊢ Fun F \to F^{-1} [A \cap B] = F^{-1} [A] \cap F^{-1} [B]$