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Theorem fmufil 22856
Description: An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fmufil ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋))

Proof of Theorem fmufil
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ufilfil 22801 . . . 4 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
2 filfbas 22745 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
31, 2syl 17 . . 3 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
4 fmfil 22841 . . 3 ((𝑋𝐴𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
53, 4syl3an2 1166 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
6 simpl2 1194 . . . . . . 7 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝐿 ∈ (UFil‘𝑌))
76, 1, 23syl 18 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝐿 ∈ (fBas‘𝑌))
8 simprl 771 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
9 simpl3 1195 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝐹:𝑌𝑋)
10 simprr 773 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)
117, 8, 9, 10fmfnfm 22855 . . . . 5 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ∃𝑔 ∈ (Fil‘𝑌)(𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))
126adantr 484 . . . . . . . 8 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → 𝐿 ∈ (UFil‘𝑌))
13 simprl 771 . . . . . . . 8 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → 𝑔 ∈ (Fil‘𝑌))
14 simprrl 781 . . . . . . . 8 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → 𝐿𝑔)
15 ufilmax 22804 . . . . . . . 8 ((𝐿 ∈ (UFil‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑌) ∧ 𝐿𝑔) → 𝐿 = 𝑔)
1612, 13, 14, 15syl3anc 1373 . . . . . . 7 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → 𝐿 = 𝑔)
1716fveq2d 6721 . . . . . 6 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → ((𝑋 FilMap 𝐹)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝑔))
18 simprrr 782 . . . . . 6 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → 𝑓 = ((𝑋 FilMap 𝐹)‘𝑔))
1917, 18eqtr4d 2780 . . . . 5 ((((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) ∧ (𝑔 ∈ (Fil‘𝑌) ∧ (𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))) → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓)
2011, 19rexlimddv 3210 . . . 4 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓)
2120expr 460 . . 3 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑋)) → (((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
2221ralrimiva 3105 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∀𝑓 ∈ (Fil‘𝑋)(((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
23 isufil2 22805 . 2 (((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋) ↔ (((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋) ∧ ∀𝑓 ∈ (Fil‘𝑋)(((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓)))
245, 22, 23sylanbrc 586 1 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wss 3866  wf 6376  cfv 6380  (class class class)co 7213  fBascfbas 20351  Filcfil 22742  UFilcufil 22796   FilMap cfm 22830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1o 8202  df-er 8391  df-en 8627  df-fin 8630  df-fi 9027  df-fbas 20360  df-fg 20361  df-fil 22743  df-ufil 22798  df-fm 22835
This theorem is referenced by:  ufldom  22859  uffcfflf  22936
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