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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 9113 | . . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) |
6 | resfsupp.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
7 | 1, 2, 3, 5, 6 | ressuppfi 9132 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
8 | resfsupp.f | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
9 | funisfsupp 9111 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
10 | 8, 2, 6, 9 | syl3anc 1370 | . 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 1542 ∈ wcel 2110 ∖ cdif 3889 class class class wbr 5079 dom cdm 5590 ↾ cres 5592 Fun wfun 6426 (class class class)co 7271 supp csupp 7968 Fincfn 8716 finSupp cfsupp 9106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-supp 7969 df-1o 8288 df-en 8717 df-fin 8720 df-fsupp 9107 |
This theorem is referenced by: lincext2 45765 |
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