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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 9371 | . . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) |
6 | resfsupp.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
7 | 1, 2, 3, 5, 6 | ressuppfi 9392 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
8 | resfsupp.f | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
9 | funisfsupp 9369 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
10 | 8, 2, 6, 9 | syl3anc 1369 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
11 | 7, 10 | mpbird 256 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 ∖ cdif 3944 class class class wbr 5147 dom cdm 5675 ↾ cres 5677 Fun wfun 6536 (class class class)co 7411 supp csupp 8148 Fincfn 8941 finSupp cfsupp 9363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-supp 8149 df-1o 8468 df-en 8942 df-fin 8945 df-fsupp 9364 |
This theorem is referenced by: lincext2 47223 |
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