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Theorem ssufl 21844
Description: If 𝑌 is a subset of 𝑋 and filters extend to ultrafilters in 𝑋, then they still do in 𝑌. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
ssufl ((𝑋 ∈ UFL ∧ 𝑌𝑋) → 𝑌 ∈ UFL)

Proof of Theorem ssufl
Dummy variables 𝑓 𝑔 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 807 . . . . 5 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑋 ∈ UFL)
2 filfbas 21774 . . . . . . . 8 (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
32adantl 473 . . . . . . 7 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑌))
4 filsspw 21777 . . . . . . . . 9 (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
54adantl 473 . . . . . . . 8 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑌)
6 simplr 809 . . . . . . . . 9 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑌𝑋)
7 sspwb 5022 . . . . . . . . 9 (𝑌𝑋 ↔ 𝒫 𝑌 ⊆ 𝒫 𝑋)
86, 7sylib 208 . . . . . . . 8 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
95, 8sstrd 3719 . . . . . . 7 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑋)
10 fbasweak 21791 . . . . . . 7 ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋𝑋 ∈ UFL) → 𝑓 ∈ (fBas‘𝑋))
113, 9, 1, 10syl3anc 1439 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑋))
12 fgcl 21804 . . . . . 6 (𝑓 ∈ (fBas‘𝑋) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
1311, 12syl 17 . . . . 5 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
14 ufli 21840 . . . . 5 ((𝑋 ∈ UFL ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
151, 13, 14syl2anc 696 . . . 4 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
16 ssfg 21798 . . . . . . . . . 10 (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓))
1711, 16syl 17 . . . . . . . . 9 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ (𝑋filGen𝑓))
1817adantr 472 . . . . . . . 8 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑋filGen𝑓))
19 simprr 813 . . . . . . . 8 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑋filGen𝑓) ⊆ 𝑢)
2018, 19sstrd 3719 . . . . . . 7 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓𝑢)
21 filtop 21781 . . . . . . . 8 (𝑓 ∈ (Fil‘𝑌) → 𝑌𝑓)
2221ad2antlr 765 . . . . . . 7 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌𝑓)
2320, 22sseldd 3710 . . . . . 6 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌𝑢)
24 simprl 811 . . . . . . 7 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑢 ∈ (UFil‘𝑋))
256adantr 472 . . . . . . 7 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌𝑋)
26 trufil 21836 . . . . . . 7 ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑌𝑋) → ((𝑢t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌𝑢))
2724, 25, 26syl2anc 696 . . . . . 6 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ((𝑢t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌𝑢))
2823, 27mpbird 247 . . . . 5 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑢t 𝑌) ∈ (UFil‘𝑌))
295adantr 472 . . . . . . 7 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝒫 𝑌)
30 restid2 16214 . . . . . . 7 ((𝑌𝑓𝑓 ⊆ 𝒫 𝑌) → (𝑓t 𝑌) = 𝑓)
3122, 29, 30syl2anc 696 . . . . . 6 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓t 𝑌) = 𝑓)
32 ssrest 21103 . . . . . . 7 ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑓𝑢) → (𝑓t 𝑌) ⊆ (𝑢t 𝑌))
3324, 20, 32syl2anc 696 . . . . . 6 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓t 𝑌) ⊆ (𝑢t 𝑌))
3431, 33eqsstr3d 3746 . . . . 5 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑢t 𝑌))
35 sseq2 3733 . . . . . 6 (𝑔 = (𝑢t 𝑌) → (𝑓𝑔𝑓 ⊆ (𝑢t 𝑌)))
3635rspcev 3413 . . . . 5 (((𝑢t 𝑌) ∈ (UFil‘𝑌) ∧ 𝑓 ⊆ (𝑢t 𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓𝑔)
3728, 34, 36syl2anc 696 . . . 4 ((((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓𝑔)
3815, 37rexlimddv 3137 . . 3 (((𝑋 ∈ UFL ∧ 𝑌𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓𝑔)
3938ralrimiva 3068 . 2 ((𝑋 ∈ UFL ∧ 𝑌𝑋) → ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓𝑔)
40 ssexg 4912 . . . 4 ((𝑌𝑋𝑋 ∈ UFL) → 𝑌 ∈ V)
4140ancoms 468 . . 3 ((𝑋 ∈ UFL ∧ 𝑌𝑋) → 𝑌 ∈ V)
42 isufl 21839 . . 3 (𝑌 ∈ V → (𝑌 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓𝑔))
4341, 42syl 17 . 2 ((𝑋 ∈ UFL ∧ 𝑌𝑋) → (𝑌 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓𝑔))
4439, 43mpbird 247 1 ((𝑋 ∈ UFL ∧ 𝑌𝑋) → 𝑌 ∈ UFL)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  Vcvv 3304  wss 3680  𝒫 cpw 4266  cfv 6001  (class class class)co 6765  t crest 16204  fBascfbas 19857  filGencfg 19858  Filcfil 21771  UFilcufil 21825  UFLcufl 21826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-rest 16206  df-fbas 19866  df-fg 19867  df-fil 21772  df-ufil 21827  df-ufl 21828
This theorem is referenced by:  ufldom  21888
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