Theorem trfilss 21693

Description: If A is a member of the filter, then the filter truncated to A is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
trfilss	|- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ( F ↾t A ) C_ F )

Proof of Theorem trfilss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		restval 16087	. 2 |- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ( F ↾t A ) = ran ( x e. F |-> ( x i^i A ) ) )
2		filin 21658	. . . . . 6 |- ( ( F e. ( Fil ` X ) /\ x e. F /\ A e. F ) -> ( x i^i A ) e. F )
3	2	3expa 1265	. . . . 5 |- ( ( ( F e. ( Fil ` X ) /\ x e. F ) /\ A e. F ) -> ( x i^i A ) e. F )
4	3	an32s 846	. . . 4 |- ( ( ( F e. ( Fil ` X ) /\ A e. F ) /\ x e. F ) -> ( x i^i A ) e. F )
5		eqid 2622	. . . 4 |- ( x e. F |-> ( x i^i A ) ) = ( x e. F |-> ( x i^i A ) )
6	4, 5	fmptd 6385	. . 3 |- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ( x e. F |-> ( x i^i A ) ) : F --> F )
7		frn 6053	. . 3 |- ( ( x e. F |-> ( x i^i A ) ) : F --> F -> ran ( x e. F |-> ( x i^i A ) ) C_ F )
8	6, 7	syl 17	. 2 |- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ran ( x e. F |-> ( x i^i A ) ) C_ F )
9	1, 8	eqsstrd 3639	1 |- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ( F ↾t A ) C_ F )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Filcfil 21649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083  df-fbas 19743  df-fil 21650

This theorem is referenced by:  fgtr  21694  flimrest  21787