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Theorem fmval 17980
 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 natural number 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
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem fmval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fm 17975 . . . . 5
21a1i 11 . . . 4
3 dmeq 5073 . . . . . . . 8
43fveq2d 5735 . . . . . . 7
54adantl 454 . . . . . 6
6 id 21 . . . . . . 7
7 imaeq1 5201 . . . . . . . . 9
87mpteq2dv 4299 . . . . . . . 8
98rneqd 5100 . . . . . . 7
106, 9oveqan12d 6103 . . . . . 6
115, 10mpteq12dv 4290 . . . . 5
12 fdm 5598 . . . . . . . 8
1312fveq2d 5735 . . . . . . 7
1413mpteq1d 4293 . . . . . 6
15143ad2ant3 981 . . . . 5
1611, 15sylan9eqr 2492 . . . 4
17 elex 2966 . . . . 5
18173ad2ant1 979 . . . 4
19 simp3 960 . . . . 5
20 elfvdm 5760 . . . . . 6
21203ad2ant2 980 . . . . 5
22 simp1 958 . . . . 5
23 fex2 5606 . . . . 5
2419, 21, 22, 23syl3anc 1185 . . . 4
25 fvex 5745 . . . . . 6
2625mptex 5969 . . . . 5
2726a1i 11 . . . 4
282, 16, 18, 24, 27ovmpt2d 6204 . . 3
2928fveq1d 5733 . 2
30 mpteq1 4292 . . . . . 6
3130rneqd 5100 . . . . 5
3231oveq2d 6100 . . . 4
33 eqid 2438 . . . 4
34 ovex 6109 . . . 4
3532, 33, 34fvmpt 5809 . . 3