Theorem cvmopn

Description: A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015)

	Ref	Expression
Assertion	cvmopn	⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝐴 ∈ 𝐶 ) → ( 𝐹 “ 𝐴 ) ∈ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } ) = ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } )
2		eqid	⊢ ∪ 𝐶 = ∪ 𝐶
3	1 2	cvmopnlem	⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝐴 ∈ 𝐶 ) → ( 𝐹 “ 𝐴 ) ∈ 𝐽 )