Theorem cvmsuni

Description: An even covering of U has union equal to the preimage of U by F . (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Hypothesis	cvmcov.1	⊢ 𝑆 = ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } )
Assertion	cvmsuni	⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) → ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		cvmcov.1	⊢ 𝑆 = ( 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	1	cvmsi	⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) → ( 𝑈 ∈ 𝐽 ∧ ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) )
3	2	simp3d	⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) → ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) )
4	3	simpld	⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) → ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) )