Metamath Proof Explorer


Theorem conncompconn

Description: The connected component containing A is connected. (Contributed by Mario Carneiro, 19-Mar-2015)

Ref Expression
Hypothesis conncomp.2 𝑆 = { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) }
Assertion conncompconn ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) → ( 𝐽t 𝑆 ) ∈ Conn )

Proof

Step Hyp Ref Expression
1 conncomp.2 𝑆 = { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) }
2 uniiun { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } = 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } 𝑦
3 1 2 eqtri 𝑆 = 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } 𝑦
4 3 oveq2i ( 𝐽t 𝑆 ) = ( 𝐽t 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } 𝑦 )
5 simpl ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
6 simpr ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } )
7 eleq2w ( 𝑥 = 𝑦 → ( 𝐴𝑥𝐴𝑦 ) )
8 oveq2 ( 𝑥 = 𝑦 → ( 𝐽t 𝑥 ) = ( 𝐽t 𝑦 ) )
9 8 eleq1d ( 𝑥 = 𝑦 → ( ( 𝐽t 𝑥 ) ∈ Conn ↔ ( 𝐽t 𝑦 ) ∈ Conn ) )
10 7 9 anbi12d ( 𝑥 = 𝑦 → ( ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) ↔ ( 𝐴𝑦 ∧ ( 𝐽t 𝑦 ) ∈ Conn ) ) )
11 10 elrab ( 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ↔ ( 𝑦 ∈ 𝒫 𝑋 ∧ ( 𝐴𝑦 ∧ ( 𝐽t 𝑦 ) ∈ Conn ) ) )
12 6 11 sylib ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → ( 𝑦 ∈ 𝒫 𝑋 ∧ ( 𝐴𝑦 ∧ ( 𝐽t 𝑦 ) ∈ Conn ) ) )
13 12 simpld ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → 𝑦 ∈ 𝒫 𝑋 )
14 13 elpwid ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → 𝑦𝑋 )
15 12 simprd ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → ( 𝐴𝑦 ∧ ( 𝐽t 𝑦 ) ∈ Conn ) )
16 15 simpld ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → 𝐴𝑦 )
17 15 simprd ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) ∧ 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } ) → ( 𝐽t 𝑦 ) ∈ Conn )
18 5 14 16 17 iunconn ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) → ( 𝐽t 𝑦 ∈ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝐴𝑥 ∧ ( 𝐽t 𝑥 ) ∈ Conn ) } 𝑦 ) ∈ Conn )
19 4 18 eqeltrid ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴𝑋 ) → ( 𝐽t 𝑆 ) ∈ Conn )