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Theorem fmval 23966
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 23961 . . . . 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 5916 . . . . . . . 8 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43fveq2d 6910 . . . . . . 7 (𝑓 = 𝐹 → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
54adantl 481 . . . . . 6 ((𝑥 = 𝑋𝑓 = 𝐹) → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
6 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
7 imaeq1 6074 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
87mpteq2dv 5249 . . . . . . . 8 (𝑓 = 𝐹 → (𝑦𝑏 ↦ (𝑓𝑦)) = (𝑦𝑏 ↦ (𝐹𝑦)))
98rneqd 5951 . . . . . . 7 (𝑓 = 𝐹 → ran (𝑦𝑏 ↦ (𝑓𝑦)) = ran (𝑦𝑏 ↦ (𝐹𝑦)))
106, 9oveqan12d 7449 . . . . . 6 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
115, 10mpteq12dv 5238 . . . . 5 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
12 fdm 6745 . . . . . . . 8 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
1312fveq2d 6910 . . . . . . 7 (𝐹:𝑌𝑋 → (fBas‘dom 𝐹) = (fBas‘𝑌))
1413mpteq1d 5242 . . . . . 6 (𝐹:𝑌𝑋 → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
15143ad2ant3 1134 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
1611, 15sylan9eqr 2796 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
17 elex 3498 . . . . 5 (𝑋𝐴𝑋 ∈ V)
18173ad2ant1 1132 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋 ∈ V)
19 simp3 1137 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
20 elfvdm 6943 . . . . . 6 (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
21203ad2ant2 1133 . . . . 5 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑌 ∈ dom fBas)
2219, 21fexd 7246 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ V)
23 fvex 6919 . . . . . 6 (fBas‘𝑌) ∈ V
2423mptex 7242 . . . . 5 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V
2524a1i 11 . . . 4 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V)
262, 16, 18, 22, 25ovmpod 7584 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
2726fveq1d 6908 . 2 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))‘𝐵))
28 mpteq1 5240 . . . . . 6 (𝑏 = 𝐵 → (𝑦𝑏 ↦ (𝐹𝑦)) = (𝑦𝐵 ↦ (𝐹𝑦)))
2928rneqd 5951 . . . . 5 (𝑏 = 𝐵 → ran (𝑦𝑏 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦)))
3029oveq2d 7446 . . . 4 (𝑏 = 𝐵 → (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
31 eqid 2734 . . . 4 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
32 ovex 7463 . . . 4 (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ∈ V
3330, 31, 32fvmpt 7015 . . 3 (𝐵 ∈ (fBas‘𝑌) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
34333ad2ant2 1133 . 2 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
3527, 34eqtrd 2774 1 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  cmpt 5230  dom cdm 5688  ran crn 5689  cima 5691  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  fBascfbas 21369  filGencfg 21370   FilMap cfm 23956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-fm 23961
This theorem is referenced by:  fmfil  23967  fmss  23969  elfm  23970  ucnextcn  24328  fmcfil  25319
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