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Mirrors > Home > MPE Home > Th. List > trfilss | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
trfilss | ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restval 16692 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) = ran (𝑥 ∈ 𝐹 ↦ (𝑥 ∩ 𝐴))) | |
2 | filin 22459 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝐴 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ 𝐹) | |
3 | 2 | 3expa 1115 | . . . . 5 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ 𝐹) |
4 | 3 | an32s 651 | . . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ 𝐹) |
5 | 4 | fmpttd 6856 | . . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 ↦ (𝑥 ∩ 𝐴)):𝐹⟶𝐹) |
6 | 5 | frnd 6494 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ran (𝑥 ∈ 𝐹 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝐹) |
7 | 1, 6 | eqsstrd 3953 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ↾t crest 16686 Filcfil 22450 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rest 16688 df-fbas 20088 df-fil 22451 |
This theorem is referenced by: fgtr 22495 flimrest 22588 |
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