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Theorem trfilss 22070
 Description: If 𝐴 is a member of the filter, then the filter truncated to 𝐴 is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
trfilss ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)

Proof of Theorem trfilss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 restval 16447 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) = ran (𝑥𝐹 ↦ (𝑥𝐴)))
2 filin 22035 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝐴𝐹) → (𝑥𝐴) ∈ 𝐹)
323expa 1151 . . . . 5 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) ∧ 𝐴𝐹) → (𝑥𝐴) ∈ 𝐹)
43an32s 642 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → (𝑥𝐴) ∈ 𝐹)
54fmpttd 6639 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 ↦ (𝑥𝐴)):𝐹𝐹)
65frnd 6289 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ran (𝑥𝐹 ↦ (𝑥𝐴)) ⊆ 𝐹)
71, 6eqsstrd 3864 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2164   ∩ cin 3797   ⊆ wss 3798   ↦ cmpt 4954  ran crn 5347  ‘cfv 6127  (class class class)co 6910   ↾t crest 16441  Filcfil 22026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-rest 16443  df-fbas 20110  df-fil 22027 This theorem is referenced by:  fgtr  22071  flimrest  22164
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