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Theorem fmval 20736
 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
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem fmval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fm 20731 . . . . 5
21a1i 11 . . . 4
3 dmeq 5024 . . . . . . . 8
43fveq2d 5853 . . . . . . 7
54adantl 464 . . . . . 6
6 id 22 . . . . . . 7
7 imaeq1 5152 . . . . . . . . 9
87mpteq2dv 4482 . . . . . . . 8
98rneqd 5051 . . . . . . 7
106, 9oveqan12d 6297 . . . . . 6
115, 10mpteq12dv 4473 . . . . 5
12 fdm 5718 . . . . . . . 8
1312fveq2d 5853 . . . . . . 7
1413mpteq1d 4476 . . . . . 6
15143ad2ant3 1020 . . . . 5
1611, 15sylan9eqr 2465 . . . 4
17 elex 3068 . . . . 5
18173ad2ant1 1018 . . . 4
19 simp3 999 . . . . 5
20 elfvdm 5875 . . . . . 6
21203ad2ant2 1019 . . . . 5
22 simp1 997 . . . . 5
23 fex2 6739 . . . . 5
2419, 21, 22, 23syl3anc 1230 . . . 4
25 fvex 5859 . . . . . 6
2625mptex 6124 . . . . 5
2726a1i 11 . . . 4
282, 16, 18, 24, 27ovmpt2d 6411 . . 3
2928fveq1d 5851 . 2
30 mpteq1 4475 . . . . . 6
3130rneqd 5051 . . . . 5
3231oveq2d 6294 . . . 4
33 eqid 2402 . . . 4
34 ovex 6306 . . . 4
3532, 33, 34fvmpt 5932 . . 3
36353ad2ant2 1019 . 2
3729, 36eqtrd 2443 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  cvv 3059   cmpt 4453   cdm 4823   crn 4824  cima 4826  wf 5565  cfv 5569  (class class class)co 6278   cmpt2 6280  cfbas 18726  cfg 18727   cfm 20726 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fm 20731 This theorem is referenced by:  fmfil  20737  fmss  20739  elfm  20740  ucnextcn  21099  fmcfil  22003
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