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Theorem fmufil 22574
 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 22519 . . . 4 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
2 filfbas 22463 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
31, 2syl 17 . . 3 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
4 fmfil 22559 . . 3 ((𝑋𝐴𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
53, 4syl3an2 1161 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
6 simpl2 1189 . . . . . . 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 770 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
9 simpl3 1190 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝐹:𝑌𝑋)
10 simprr 772 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)
117, 8, 9, 10fmfnfm 22573 . . . . 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 770 . . . . . . . 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 780 . . . . . . . 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 22522 . . . . . . . 8 ((𝐿 ∈ (UFil‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑌) ∧ 𝐿𝑔) → 𝐿 = 𝑔)
1612, 13, 14, 15syl3anc 1368 . . . . . . 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 6650 . . . . . 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 781 . . . . . 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 2836 . . . . 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 3250 . . . 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 3149 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∀𝑓 ∈ (Fil‘𝑋)(((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
23 isufil2 22523 . 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 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  fBascfbas 20083  Filcfil 22460  UFilcufil 22514   FilMap cfm 22548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-fin 8499  df-fi 8862  df-fbas 20092  df-fg 20093  df-fil 22461  df-ufil 22516  df-fm 22553 This theorem is referenced by:  ufldom  22577  uffcfflf  22654
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