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Theorem fmf 21659
Description: Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fmf ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Proof of Theorem fmf
Dummy variables 𝑓 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6632 . . . 4 (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))) ∈ V
2 eqid 2621 . . . 4 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
31, 2fnmpti 5979 . . 3 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) Fn (fBas‘𝑌)
4 df-fm 21652 . . . . . 6 FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))))
54a1i 11 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))))))
6 dmeq 5284 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
76adantl 482 . . . . . . . 8 ((𝑥 = 𝑋𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
8 fdm 6008 . . . . . . . . 9 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
983ad2ant3 1082 . . . . . . . 8 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → dom 𝐹 = 𝑌)
107, 9sylan9eqr 2677 . . . . . . 7 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → dom 𝑓 = 𝑌)
1110fveq2d 6152 . . . . . 6 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (fBas‘dom 𝑓) = (fBas‘𝑌))
12 id 22 . . . . . . . 8 (𝑥 = 𝑋𝑥 = 𝑋)
13 imaeq1 5420 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1413mpteq2dv 4705 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑦𝑏 ↦ (𝑓𝑦)) = (𝑦𝑏 ↦ (𝐹𝑦)))
1514rneqd 5313 . . . . . . . 8 (𝑓 = 𝐹 → ran (𝑦𝑏 ↦ (𝑓𝑦)) = ran (𝑦𝑏 ↦ (𝐹𝑦)))
1612, 15oveqan12d 6623 . . . . . . 7 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
1716adantl 482 . . . . . 6 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
1811, 17mpteq12dv 4693 . . . . 5 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
19 elex 3198 . . . . . 6 (𝑋𝐴𝑋 ∈ V)
20193ad2ant1 1080 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → 𝑋 ∈ V)
21 fex2 7068 . . . . . 6 ((𝐹:𝑌𝑋𝑌𝐵𝑋𝐴) → 𝐹 ∈ V)
22213com13 1267 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → 𝐹 ∈ V)
23 fvex 6158 . . . . . . 7 (fBas‘𝑌) ∈ V
2423mptex 6440 . . . . . 6 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V
2524a1i 11 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) ∈ V)
265, 18, 20, 22, 25ovmpt2d 6741 . . . 4 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
2726fneq1d 5939 . . 3 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ↔ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) Fn (fBas‘𝑌)))
283, 27mpbiri 248 . 2 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
29 simpl1 1062 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐴)
30 simpr 477 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
31 simpl3 1064 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
32 fmfil 21658 . . . 4 ((𝑋𝐴𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
3329, 30, 31, 32syl3anc 1323 . . 3 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
3433ralrimiva 2960 . 2 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
35 ffnfv 6343 . 2 ((𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋) ↔ ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ∧ ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋)))
3628, 34, 35sylanbrc 697 1 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cmpt 4673  dom cdm 5074  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  fBascfbas 19653  filGencfg 19654  Filcfil 21559   FilMap cfm 21647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-fbas 19662  df-fg 19663  df-fil 21560  df-fm 21652
This theorem is referenced by:  rnelfm  21667
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