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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 8557 | . . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) |
6 | resfsupp.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
7 | 1, 2, 3, 5, 6 | ressuppfi 8576 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
8 | resfsupp.f | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
9 | funisfsupp 8555 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
10 | 8, 2, 6, 9 | syl3anc 1494 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
11 | 7, 10 | mpbird 249 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ∖ cdif 3795 class class class wbr 4875 dom cdm 5346 ↾ cres 5348 Fun wfun 6121 (class class class)co 6910 supp csupp 7564 Fincfn 8228 finSupp cfsupp 8550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-oadd 7835 df-er 8014 df-en 8229 df-fin 8232 df-fsupp 8551 |
This theorem is referenced by: lincext2 43105 |
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