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Theorem fmval 22548
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 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
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.
StepHypRef Expression
1 df-fm 22543 . . . . 5 FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))))
21a1i 11 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))))))
3 dmeq 5736 . . . . . . . 8 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43fveq2d 6649 . . . . . . 7 (𝑓 = 𝐹 → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
54adantl 485 . . . . . 6 ((𝑥 = 𝑋𝑓 = 𝐹) → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
6 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
7 imaeq1 5891 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
87mpteq2dv 5126 . . . . . . . 8 (𝑓 = 𝐹 → (𝑦𝑏 ↦ (𝑓𝑦)) = (𝑦𝑏 ↦ (𝐹𝑦)))
98rneqd 5772 . . . . . . 7 (𝑓 = 𝐹 → ran (𝑦𝑏 ↦ (𝑓𝑦)) = ran (𝑦𝑏 ↦ (𝐹𝑦)))
106, 9oveqan12d 7154 . . . . . 6 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
115, 10mpteq12dv 5115 . . . . 5 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
12 fdm 6495 . . . . . . . 8 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
1312fveq2d 6649 . . . . . . 7 (𝐹:𝑌𝑋 → (fBas‘dom 𝐹) = (fBas‘𝑌))
1413mpteq1d 5119 . . . . . 6 (𝐹:𝑌𝑋 → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
15143ad2ant3 1132 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
1611, 15sylan9eqr 2855 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
17 elex 3459 . . . . 5 (𝑋𝐴𝑋 ∈ V)
18173ad2ant1 1130 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋 ∈ V)
19 simp3 1135 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
20 elfvdm 6677 . . . . . 6 (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
21203ad2ant2 1131 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ dom fBas)
22 simp1 1133 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋𝐴)
23 fex2 7620 . . . . 5 ((𝐹:𝑌𝑋𝑌 ∈ dom fBas ∧ 𝑋𝐴) → 𝐹 ∈ V)
2419, 21, 22, 23syl3anc 1368 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ V)
25 fvex 6658 . . . . . 6 (fBas‘𝑌) ∈ V
2625mptex 6963 . . . . 5 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V
2726a1i 11 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V)
282, 16, 18, 24, 27ovmpod 7281 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
2928fveq1d 6647 . 2 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))‘𝐵))
30 mpteq1 5118 . . . . . 6 (𝑏 = 𝐵 → (𝑦𝑏 ↦ (𝐹𝑦)) = (𝑦𝐵 ↦ (𝐹𝑦)))
3130rneqd 5772 . . . . 5 (𝑏 = 𝐵 → ran (𝑦𝑏 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦)))
3231oveq2d 7151 . . . 4 (𝑏 = 𝐵 → (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
33 eqid 2798 . . . 4 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
34 ovex 7168 . . . 4 (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ∈ V
3532, 33, 34fvmpt 6745 . . 3 (𝐵 ∈ (fBas‘𝑌) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
36353ad2ant2 1131 . 2 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
3729, 36eqtrd 2833 1 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  Vcvv 3441  cmpt 5110  dom cdm 5519  ran crn 5520  cima 5522  wf 6320  cfv 6324  (class class class)co 7135  cmpo 7137  fBascfbas 20079  filGencfg 20080   FilMap cfm 22538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-fm 22543
This theorem is referenced by:  fmfil  22549  fmss  22551  elfm  22552  ucnextcn  22910  fmcfil  23876
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