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Theorem fmufil 23685
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 23630 . . . 4 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
2 filfbas 23574 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
31, 2syl 17 . . 3 (𝐿 ∈ (UFil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
4 fmfil 23670 . . 3 ((𝑋𝐴𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
53, 4syl3an2 1162 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
6 simpl2 1190 . . . . . . 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 767 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
9 simpl3 1191 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → 𝐹:𝑌𝑋)
10 simprr 769 . . . . . 6 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)
117, 8, 9, 10fmfnfm 23684 . . . . 5 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ∃𝑔 ∈ (Fil‘𝑌)(𝐿𝑔𝑓 = ((𝑋 FilMap 𝐹)‘𝑔)))
126adantr 479 . . . . . . . 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 767 . . . . . . . 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 777 . . . . . . . 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 23633 . . . . . . . 8 ((𝐿 ∈ (UFil‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑌) ∧ 𝐿𝑔) → 𝐿 = 𝑔)
1612, 13, 14, 15syl3anc 1369 . . . . . . 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 6896 . . . . . 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 778 . . . . . 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 2773 . . . . 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 3159 . . . 4 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓)) → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓)
2120expr 455 . . 3 (((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑋)) → (((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
2221ralrimiva 3144 . 2 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∀𝑓 ∈ (Fil‘𝑋)(((𝑋 FilMap 𝐹)‘𝐿) ⊆ 𝑓 → ((𝑋 FilMap 𝐹)‘𝐿) = 𝑓))
23 isufil2 23634 . 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 581 1 ((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wss 3949  wf 6540  cfv 6544  (class class class)co 7413  fBascfbas 21134  Filcfil 23571  UFilcufil 23625   FilMap cfm 23659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1o 8470  df-er 8707  df-en 8944  df-fin 8947  df-fi 9410  df-fbas 21143  df-fg 21144  df-fil 23572  df-ufil 23627  df-fm 23664
This theorem is referenced by:  ufldom  23688  uffcfflf  23765
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