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Metamath Proof Explorer

Theorem fmid

Description: The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009) (Revised by Mario Carneiro, 27-Aug-2015)

Ref Expression
Assertion fmid ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ( 𝑋 FilMap ( I ↾ 𝑋 ) ) ‘ 𝐹 ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 filfbas ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )
2 f1oi ⊢ ( I ↾ 𝑋 ) : 𝑋 –1-1-onto→ 𝑋
3 f1ofo ⊢ ( ( I ↾ 𝑋 ) : 𝑋 –1-1-onto→ 𝑋 → ( I ↾ 𝑋 ) : 𝑋 –onto→ 𝑋 )
4 2 3 ax-mp ⊢ ( I ↾ 𝑋 ) : 𝑋 –onto→ 𝑋
5 eqid ⊢ ( 𝑋 filGen 𝐹 ) = ( 𝑋 filGen 𝐹 )
6 5 elfm3 ⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ ( I ↾ 𝑋 ) : 𝑋 –onto→ 𝑋 ) → ( 𝑡 ∈ ( ( 𝑋 FilMap ( I ↾ 𝑋 ) ) ‘ 𝐹 ) ↔ ∃ 𝑠 ∈ ( 𝑋 filGen 𝐹 ) 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ) )
7 1 4 6 sylancl ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑡 ∈ ( ( 𝑋 FilMap ( I ↾ 𝑋 ) ) ‘ 𝐹 ) ↔ ∃ 𝑠 ∈ ( 𝑋 filGen 𝐹 ) 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ) )
8 fgfil ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑋 filGen 𝐹 ) = 𝐹 )
9 8 rexeqdv ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∃ 𝑠 ∈ ( 𝑋 filGen 𝐹 ) 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ↔ ∃ 𝑠 ∈ 𝐹 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ) )
10 filelss ⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝑠 ⊆ 𝑋 )
11 resiima ⊢ ( 𝑠 ⊆ 𝑋 → ( ( I ↾ 𝑋 ) “ 𝑠 ) = 𝑠 )
12 10 11 syl ⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → ( ( I ↾ 𝑋 ) “ 𝑠 ) = 𝑠 )
13 12 eqeq2d ⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → ( 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ↔ 𝑡 = 𝑠 ) )
14 equcom ⊢ ( 𝑠 = 𝑡 ↔ 𝑡 = 𝑠 )
15 13 14 bitr4di ⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → ( 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ↔ 𝑠 = 𝑡 ) )
16 15 rexbidva ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∃ 𝑠 ∈ 𝐹 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ↔ ∃ 𝑠 ∈ 𝐹 𝑠 = 𝑡 ) )
17 risset ⊢ ( 𝑡 ∈ 𝐹 ↔ ∃ 𝑠 ∈ 𝐹 𝑠 = 𝑡 )
18 16 17 bitr4di ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∃ 𝑠 ∈ 𝐹 𝑡 = ( ( I ↾ 𝑋 ) “ 𝑠 ) ↔ 𝑡 ∈ 𝐹 ) )
19 7 9 18 3bitrd ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( 𝑡 ∈ ( ( 𝑋 FilMap ( I ↾ 𝑋 ) ) ‘ 𝐹 ) ↔ 𝑡 ∈ 𝐹 ) )
20 19 eqrdv ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ( 𝑋 FilMap ( I ↾ 𝑋 ) ) ‘ 𝐹 ) = 𝐹 )