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Mirrors > Home > MPE Home > Th. List > resfsupp | Structured version Visualization version GIF version |
Description: If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.) |
Ref | Expression |
---|---|
resfsupp.b | ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) |
resfsupp.e | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
resfsupp.f | ⊢ (𝜑 → Fun 𝐹) |
resfsupp.g | ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵)) |
resfsupp.s | ⊢ (𝜑 → 𝐺 finSupp 𝑍) |
resfsupp.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
resfsupp | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resfsupp.b | . . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) | |
2 | resfsupp.e | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
3 | resfsupp.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵)) | |
4 | resfsupp.s | . . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
5 | 4 | fsuppimpd 8828 | . . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) |
6 | resfsupp.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
7 | 1, 2, 3, 5, 6 | ressuppfi 8847 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
8 | resfsupp.f | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
9 | funisfsupp 8826 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
10 | 8, 2, 6, 9 | syl3anc 1363 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
11 | 7, 10 | mpbird 258 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 class class class wbr 5057 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 (class class class)co 7145 supp csupp 7819 Fincfn 8497 finSupp cfsupp 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-oadd 8095 df-er 8278 df-en 8498 df-fin 8501 df-fsupp 8822 |
This theorem is referenced by: lincext2 44438 |
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