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 )