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Theorem fmufil 21964
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 21909 . . . 4 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
2 filfbas 21853 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
31, 2syl 17 . . 3 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
4 fmfil 21949 . . 3 ((𝑋𝐴𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
53, 4syl3an2 1168 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
6 simpl2 1230 . . . . . . 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 811 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
9 simpl3 1232 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝐹:𝑌𝑋)
10 simprr 813 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)
117, 8, 9, 10fmfnfm 21963 . . . . 5 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ∃𝑔 ∈ (Fil‘𝑌)(𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))
126adantr 472 . . . . . . . 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 811 . . . . . . . 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 823 . . . . . . . 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 21912 . . . . . . . 8 ((𝐿 ∈ (UFil‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑌) ∧ 𝐿𝑔) → 𝐿 = 𝑔)
1612, 13, 14, 15syl3anc 1477 . . . . . . 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 6356 . . . . . 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 824 . . . . . 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 2797 . . . . 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 3173 . . . 4 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓)
2120expr 644 . . 3 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑋)) → (((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
2221ralrimiva 3104 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∀𝑓 ∈ (Fil‘𝑋)(((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
23 isufil2 21913 . 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 701 1 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wss 3715  wf 6045  cfv 6049  (class class class)co 6813  fBascfbas 19936  Filcfil 21850  UFilcufil 21904   FilMap cfm 21938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-fbas 19945  df-fg 19946  df-fil 21851  df-ufil 21906  df-fm 21943
This theorem is referenced by:  ufldom  21967  uffcfflf  22044
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