Theorem fmval 21747

Description: Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual positive integer ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
fmval	|- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( X filGen ran ( y e. B |-> ( F " y ) ) ) )

Distinct variable groups:   y, B   y, F   y, X   y, Y   y, A

Proof of Theorem fmval

Dummy variables f b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fm 21742	. . . . 5 |- FilMap = ( x e. _V , f e. _V |-> ( b e. ( fBas ` dom f ) |-> ( x filGen ran ( y e. b |-> ( f " y ) ) ) ) )
2	1	a1i 11	. . . 4 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> FilMap = ( x e. _V , f e. _V |-> ( b e. ( fBas ` dom f ) |-> ( x filGen ran ( y e. b |-> ( f " y ) ) ) ) ) )
3		dmeq 5324	. . . . . . . 8 |- ( f = F -> dom f = dom F )
4	3	fveq2d 6195	. . . . . . 7 |- ( f = F -> ( fBas ` dom f ) = ( fBas ` dom F ) )
5	4	adantl 482	. . . . . 6 |- ( ( x = X /\ f = F ) -> ( fBas ` dom f ) = ( fBas ` dom F ) )
6		id 22	. . . . . . 7 |- ( x = X -> x = X )
7		imaeq1 5461	. . . . . . . . 9 |- ( f = F -> ( f " y ) = ( F " y ) )
8	7	mpteq2dv 4745	. . . . . . . 8 |- ( f = F -> ( y e. b |-> ( f " y ) ) = ( y e. b |-> ( F " y ) ) )
9	8	rneqd 5353	. . . . . . 7 |- ( f = F -> ran ( y e. b |-> ( f " y ) ) = ran ( y e. b |-> ( F " y ) ) )
10	6, 9	oveqan12d 6669	. . . . . 6 |- ( ( x = X /\ f = F ) -> ( x filGen ran ( y e. b |-> ( f " y ) ) ) = ( X filGen ran ( y e. b |-> ( F " y ) ) ) )
11	5, 10	mpteq12dv 4733	. . . . 5 |- ( ( x = X /\ f = F ) -> ( b e. ( fBas ` dom f ) |-> ( x filGen ran ( y e. b |-> ( f " y ) ) ) ) = ( b e. ( fBas ` dom F ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) )
12		fdm 6051	. . . . . . . 8 |- ( F : Y --> X -> dom F = Y )
13	12	fveq2d 6195	. . . . . . 7 |- ( F : Y --> X -> ( fBas ` dom F ) = ( fBas ` Y ) )
14	13	mpteq1d 4738	. . . . . 6 |- ( F : Y --> X -> ( b e. ( fBas ` dom F ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) = ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) )
15	14	3ad2ant3 1084	. . . . 5 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( b e. ( fBas ` dom F ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) = ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) )
16	11, 15	sylan9eqr 2678	. . . 4 |- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ ( x = X /\ f = F ) ) -> ( b e. ( fBas ` dom f ) |-> ( x filGen ran ( y e. b |-> ( f " y ) ) ) ) = ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) )
17		elex 3212	. . . . 5 |- ( X e. A -> X e. _V )
18	17	3ad2ant1 1082	. . . 4 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> X e. _V )
19		simp3 1063	. . . . 5 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> F : Y --> X )
20		elfvdm 6220	. . . . . 6 |- ( B e. ( fBas ` Y ) -> Y e. dom fBas )
21	20	3ad2ant2 1083	. . . . 5 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> Y e. dom fBas )
22		simp1 1061	. . . . 5 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> X e. A )
23		fex2 7121	. . . . 5 |- ( ( F : Y --> X /\ Y e. dom fBas /\ X e. A ) -> F e. _V )
24	19, 21, 22, 23	syl3anc 1326	. . . 4 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> F e. _V )
25		fvex 6201	. . . . . 6 |- ( fBas ` Y ) e. _V
26	25	mptex 6486	. . . . 5 |- ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) e. _V
27	26	a1i 11	. . . 4 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) e. _V )
28	2, 16, 18, 24, 27	ovmpt2d 6788	. . 3 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( X FilMap F ) = ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) )
29	28	fveq1d 6193	. 2 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) ` B ) )
30		mpteq1 4737	. . . . . 6 |- ( b = B -> ( y e. b |-> ( F " y ) ) = ( y e. B |-> ( F " y ) ) )
31	30	rneqd 5353	. . . . 5 |- ( b = B -> ran ( y e. b |-> ( F " y ) ) = ran ( y e. B |-> ( F " y ) ) )
32	31	oveq2d 6666	. . . 4 |- ( b = B -> ( X filGen ran ( y e. b |-> ( F " y ) ) ) = ( X filGen ran ( y e. B |-> ( F " y ) ) ) )
33		eqid 2622	. . . 4 |- ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) = ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) )
34		ovex 6678	. . . 4 |- ( X filGen ran ( y e. B |-> ( F " y ) ) ) e. _V
35	32, 33, 34	fvmpt 6282	. . 3 |- ( B e. ( fBas ` Y ) -> ( ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) ` B ) = ( X filGen ran ( y e. B |-> ( F " y ) ) ) )
36	35	3ad2ant2 1083	. 2 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( b e. ( fBas ` Y ) |-> ( X filGen ran ( y e. b |-> ( F " y ) ) ) ) ` B ) = ( X filGen ran ( y e. B |-> ( F " y ) ) ) )
37	29, 36	eqtrd 2656	1 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( X filGen ran ( y e. B |-> ( F " y ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   fBascfbas 19734   filGencfg 19735   FilMap cfm 21737

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-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-fm 21742

This theorem is referenced by:  fmfil  21748  fmss  21750  elfm  21751  ucnextcn  22108  fmcfil  23070