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Theorem fmf 21969
 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 6827 . . . 4 (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))) ∈ V
2 eqid 2771 . . . 4 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
31, 2fnmpti 6161 . . 3 (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))) Fn (fBas‘𝑌)
4 df-fm 21962 . . . . . 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 5461 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
76adantl 467 . . . . . . . 8 ((𝑥 = 𝑋𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
8 fdm 6192 . . . . . . . . 9 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
983ad2ant3 1129 . . . . . . . 8 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → dom 𝐹 = 𝑌)
107, 9sylan9eqr 2827 . . . . . . 7 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → dom 𝑓 = 𝑌)
1110fveq2d 6337 . . . . . 6 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (fBas‘dom 𝑓) = (fBas‘𝑌))
12 id 22 . . . . . . . 8 (𝑥 = 𝑋𝑥 = 𝑋)
13 imaeq1 5601 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
1413mpteq2dv 4880 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑦𝑏 ↦ (𝑓𝑦)) = (𝑦𝑏 ↦ (𝐹𝑦)))
1514rneqd 5490 . . . . . . . 8 (𝑓 = 𝐹 → ran (𝑦𝑏 ↦ (𝑓𝑦)) = ran (𝑦𝑏 ↦ (𝐹𝑦)))
1612, 15oveqan12d 6815 . . . . . . 7 ((𝑥 = 𝑋𝑓 = 𝐹) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
1716adantl 467 . . . . . 6 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦))) = (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦))))
1811, 17mpteq12dv 4868 . . . . 5 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ (𝑥 = 𝑋𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦𝑏 ↦ (𝑓𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
19 elex 3364 . . . . . 6 (𝑋𝐴𝑋 ∈ V)
20193ad2ant1 1127 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → 𝑋 ∈ V)
21 fex2 7272 . . . . . 6 ((𝐹:𝑌𝑋𝑌𝐵𝑋𝐴) → 𝐹 ∈ V)
22213com13 1118 . . . . 5 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → 𝐹 ∈ V)
23 fvex 6344 . . . . . . 7 (fBas‘𝑌) ∈ V
2423mptex 6633 . . . . . 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 6939 . . . 4 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦𝑏 ↦ (𝐹𝑦)))))
2726fneq1d 6120 . . 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 1227 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐴)
30 simpr 471 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
31 simpl3 1231 . . . 4 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
32 fmfil 21968 . . . 4 ((𝑋𝐴𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
3329, 30, 31, 32syl3anc 1476 . . 3 (((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
3433ralrimiva 3115 . 2 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
35 ffnfv 6533 . 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 572 1 ((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ↦ cmpt 4864  dom cdm 5250  ran crn 5251   “ cima 5253   Fn wfn 6025  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796   ↦ cmpt2 6798  fBascfbas 19949  filGencfg 19950  Filcfil 21869   FilMap cfm 21957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-fbas 19958  df-fg 19959  df-fil 21870  df-fm 21962 This theorem is referenced by:  rnelfm  21977
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