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Theorem fmufil 23326
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 23271 . . . 4 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
2 filfbas 23215 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
31, 2syl 17 . . 3 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
4 fmfil 23311 . . 3 ((𝑋𝐴𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
53, 4syl3an2 1165 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
6 simpl2 1193 . . . . . . 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 1194 . . . . . 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 23325 . . . . 5 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ∃𝑔 ∈ (Fil‘𝑌)(𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))
126adantr 482 . . . . . . . 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 23274 . . . . . . . 8 ((𝐿 ∈ (UFil‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑌) ∧ 𝐿𝑔) → 𝐿 = 𝑔)
1612, 13, 14, 15syl3anc 1372 . . . . . . 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 6847 . . . . . 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 2776 . . . . 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 3155 . . . 4 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓)
2120expr 458 . . 3 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑋)) → (((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
2221ralrimiva 3140 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∀𝑓 ∈ (Fil‘𝑋)(((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
23 isufil2 23275 . 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 584 1 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wss 3911  wf 6493  cfv 6497  (class class class)co 7358  fBascfbas 20800  Filcfil 23212  UFilcufil 23266   FilMap cfm 23300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1o 8413  df-er 8651  df-en 8887  df-fin 8890  df-fi 9352  df-fbas 20809  df-fg 20810  df-fil 23213  df-ufil 23268  df-fm 23305
This theorem is referenced by:  ufldom  23329  uffcfflf  23406
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