Theorem t1hmph

Description: T_1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015)

	Ref	Expression
Assertion	t1hmph	$$⊢J\simeq K\to \left(J\in \mathrm{Fre}\to K\in \mathrm{Fre}\right)$$

Proof

Step	Hyp	Ref	Expression
1		t1top	$$⊢J\in \mathrm{Fre}\to J\in \mathrm{Top}$$
2		cnt1	$$⊢\left(J\in \mathrm{Fre}\wedge f:\bigcup K\underset{1-1}{⟶}\bigcup J\wedge f\in \left(K\mathrm{Cn}J\right)\right)\to K\in \mathrm{Fre}$$
3	1 2	haushmphlem	$$⊢J\simeq K\to \left(J\in \mathrm{Fre}\to K\in \mathrm{Fre}\right)$$