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Theorem trfilss 22472
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 16678 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) = ran (𝑥𝐹 ↦ (𝑥𝐴)))
2 filin 22437 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝐴𝐹) → (𝑥𝐴) ∈ 𝐹)
323expa 1115 . . . . 5 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) ∧ 𝐴𝐹) → (𝑥𝐴) ∈ 𝐹)
43an32s 651 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → (𝑥𝐴) ∈ 𝐹)
54fmpttd 6852 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 ↦ (𝑥𝐴)):𝐹𝐹)
65frnd 6494 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ran (𝑥𝐹 ↦ (𝑥𝐴)) ⊆ 𝐹)
71, 6eqsstrd 3981 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  cin 3909  wss 3910  cmpt 5119  ran crn 5529  cfv 6328  (class class class)co 7130  t crest 16672  Filcfil 22428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-rest 16674  df-fbas 20517  df-fil 22429
This theorem is referenced by:  fgtr  22473  flimrest  22566
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