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Theorem fmfil 21795
 Description: A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
fmfil ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋))

Proof of Theorem fmfil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fmval 21794 . 2 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
2 eqid 2651 . . . . 5 ran (𝑦𝐵 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦))
32fbasrn 21735 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
433comr 1290 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
5 fgcl 21729 . . 3 (ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ∈ (Fil‘𝑋))
64, 5syl 17 . 2 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ∈ (Fil‘𝑋))
71, 6eqeltrd 2730 1 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   ∈ wcel 2030   ↦ cmpt 4762  ran crn 5144   “ cima 5146  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  fBascfbas 19782  filGencfg 19783  Filcfil 21696   FilMap cfm 21784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fbas 19791  df-fg 19792  df-fil 21697  df-fm 21789 This theorem is referenced by:  fmf  21796  fmufil  21810  fmco  21812  ufldom  21813  flfnei  21842  isflf  21844  flfcnp  21855  isfcf  21885  cnpfcfi  21891  cnpfcf  21892  cnextucn  22154
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