Metamath Proof Explorer

Theorem iillysconn

Description: The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015)

		Ref	Expression
	Assertion	iillysconn	$⊢ II \in LocallySConn$

Step	Hyp	Ref	Expression
1		dfii2	$⊢ II = topGen (ran (.)) ↾_{𝑡} [0, 1]$
2		0re	$⊢ 0 \in ℝ$
3		1re	$⊢ 1 \in ℝ$
4		iccllysconn	$⊢ (0 \in ℝ \land 1 \in ℝ) \to topGen (ran (.)) ↾_{𝑡} [0, 1] \in LocallySConn$
5	2 3 4	mp2an	$⊢ topGen (ran (.)) ↾_{𝑡} [0, 1] \in LocallySConn$
6	1 5	eqeltri	$⊢ II \in LocallySConn$