iccsconn

Description: A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015)

		Ref	Expression
	Assertion	iccsconn	$⊢ (A \in ℝ \land B \in ℝ) \to topGen (ran (.)) ↾_{𝑡} [A, B] \in SConn$

Step	Hyp	Ref	Expression
1		iccconn	$⊢ (A \in ℝ \land B \in ℝ) \to topGen (ran (.)) ↾_{𝑡} [A, B] \in Conn$
2		iccssre	$⊢ (A \in ℝ \land B \in ℝ) \to [A, B] \subseteq ℝ$
3		eqid	$⊢ topGen (ran (.)) ↾_{𝑡} [A, B] = topGen (ran (.)) ↾_{𝑡} [A, B]$
4	3	resconn	$⊢ [A, B] \subseteq ℝ \to (topGen (ran (.)) ↾_{𝑡} [A, B] \in SConn \leftrightarrow topGen (ran (.)) ↾_{𝑡} [A, B] \in Conn)$
5	2 4	syl	$⊢ (A \in ℝ \land B \in ℝ) \to (topGen (ran (.)) ↾_{𝑡} [A, B] \in SConn \leftrightarrow topGen (ran (.)) ↾_{𝑡} [A, B] \in Conn)$
6	1 5	mpbird	$⊢ (A \in ℝ \land B \in ℝ) \to topGen (ran (.)) ↾_{𝑡} [A, B] \in SConn$

Metamath Proof Explorer