Metamath Proof Explorer

Theorem cvmsval

Description: Elementhood in the set S of all even coverings of an open set in J . S is an even covering of U if it is a nonempty collection of disjoint open sets in C whose union is the preimage of U , such that each set u e. S is homeomorphic under F to U . (Contributed by Mario Carneiro, 13-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 cvmsval ( 𝐶 ∈ 𝑉 → ( 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 ) ) ) ) ) ) )

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 3anass ⊢ ( ( 𝑈 ∈ 𝐽 ∧ ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 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 ) ) ) ) ) ) )
4 id ⊢ ( 𝑈 ∈ 𝐽 → 𝑈 ∈ 𝐽 )
5 pwexg ⊢ ( 𝐶 ∈ 𝑉 → 𝒫 𝐶 ∈ V )
6 difexg ⊢ ( 𝒫 𝐶 ∈ V → ( 𝒫 𝐶 ∖ { ∅ } ) ∈ V )
7 rabexg ⊢ ( ( 𝒫 𝐶 ∖ { ∅ } ) ∈ V → { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } ∈ V )
8 5 6 7 3syl ⊢ ( 𝐶 ∈ 𝑉 → { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } ∈ V )
9 imaeq2 ⊢ ( 𝑘 = 𝑈 → ( ◡ 𝐹 “ 𝑘 ) = ( ◡ 𝐹 “ 𝑈 ) )
10 9 eqeq2d ⊢ ( 𝑘 = 𝑈 → ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ↔ ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ) )
11 oveq2 ⊢ ( 𝑘 = 𝑈 → ( 𝐽 ↾t 𝑘 ) = ( 𝐽 ↾t 𝑈 ) )
12 11 oveq2d ⊢ ( 𝑘 = 𝑈 → ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) = ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) )
13 12 eleq2d ⊢ ( 𝑘 = 𝑈 → ( ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ↔ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) )
14 13 anbi2d ⊢ ( 𝑘 = 𝑈 → ( ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) )
15 14 ralbidv ⊢ ( 𝑘 = 𝑈 → ( ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) )
16 10 15 anbi12d ⊢ ( 𝑘 = 𝑈 → ( ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ↔ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) )
17 16 rabbidv ⊢ ( 𝑘 = 𝑈 → { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } = { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } )
18 17 1 fvmptg ⊢ ( ( 𝑈 ∈ 𝐽 ∧ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } ∈ V ) → ( 𝑆 ‘ 𝑈 ) = { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } )
19 4 8 18 syl2anr ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( 𝑆 ‘ 𝑈 ) = { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } )
20 19 eleq2d ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) ↔ 𝑇 ∈ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } ) )
21 unieq ⊢ ( 𝑠 = 𝑇 → ∪ 𝑠 = ∪ 𝑇 )
22 21 eqeq1d ⊢ ( 𝑠 = 𝑇 → ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ↔ ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ) )
23 difeq1 ⊢ ( 𝑠 = 𝑇 → ( 𝑠 ∖ { 𝑢 } ) = ( 𝑇 ∖ { 𝑢 } ) )
24 23 raleqdv ⊢ ( 𝑠 = 𝑇 → ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ↔ ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ) )
25 24 anbi1d ⊢ ( 𝑠 = 𝑇 → ( ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ↔ ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) )
26 25 raleqbi1dv ⊢ ( 𝑠 = 𝑇 → ( ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ↔ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) )
27 22 26 anbi12d ⊢ ( 𝑠 = 𝑇 → ( ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ↔ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) )
28 27 elrab ⊢ ( 𝑇 ∈ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } ↔ ( 𝑇 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) )
29 eldifsn ⊢ ( 𝑇 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ↔ ( 𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅ ) )
30 elpw2g ⊢ ( 𝐶 ∈ 𝑉 → ( 𝑇 ∈ 𝒫 𝐶 ↔ 𝑇 ⊆ 𝐶 ) )
31 30 adantr ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( 𝑇 ∈ 𝒫 𝐶 ↔ 𝑇 ⊆ 𝐶 ) )
32 31 anbi1d ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( ( 𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅ ) ↔ ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ) )
33 29 32 syl5bb ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( 𝑇 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ↔ ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ) )
34 33 anbi1d ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( ( 𝑇 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) ↔ ( ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) ) )
35 28 34 syl5bb ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( 𝑇 ∈ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) } ↔ ( ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) ) )
36 20 35 bitrd ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) ↔ ( ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) ) )
37 36 biimprd ⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽 ) → ( ( ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) → 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) ) )
38 37 expimpd ⊢ ( 𝐶 ∈ 𝑉 → ( ( 𝑈 ∈ 𝐽 ∧ ( ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) ) → 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) ) )
39 3 38 syl5bi ⊢ ( 𝐶 ∈ 𝑉 → ( ( 𝑈 ∈ 𝐽 ∧ ( 𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅ ) ∧ ( ∪ 𝑇 = ( ◡ 𝐹 “ 𝑈 ) ∧ ∀ 𝑢 ∈ 𝑇 ( ∀ 𝑣 ∈ ( 𝑇 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) ) ) → 𝑇 ∈ ( 𝑆 ‘ 𝑈 ) ) )
40 2 39 impbid2 ⊢ ( 𝐶 ∈ 𝑉 → ( 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 ) ) ) ) ) ) )