Theorem cvmcn

Description: A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015)

	Ref	Expression
Assertion	cvmcn	⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐹 ∈ ( 𝐶 Cn 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	⊢ ∪ 𝐽 = ∪ 𝐽
3	1 2	iscvm	⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ↔ ( ( 𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∃ 𝑘 ∈ 𝐽 ( 𝑥 ∈ 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 ) ) ) ) } ) ‘ 𝑘 ) ≠ ∅ ) ) )
4	3	simplbi	⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → ( 𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) )
5	4	simp3d	⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) )