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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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