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Mirrors > Home > MPE Home > Th. List > sniffsupp | Structured version Visualization version GIF version |
Description: A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.) |
Ref | Expression |
---|---|
sniffsupp.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
sniffsupp.0 | ⊢ (𝜑 → 0 ∈ 𝑊) |
sniffsupp.f | ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) |
Ref | Expression |
---|---|
sniffsupp | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sniffsupp.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
2 | snfi 9077 | . . . 4 ⊢ {𝑋} ∈ Fin | |
3 | eldifsni 4798 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ {𝑋}) → 𝑥 ≠ 𝑋) | |
4 | 3 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ {𝑋})) → 𝑥 ≠ 𝑋) |
5 | 4 | neneqd 2942 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ {𝑋})) → ¬ 𝑥 = 𝑋) |
6 | 5 | iffalsed 4543 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝐴, 0 ) = 0 ) |
7 | sniffsupp.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
8 | 6, 7 | suppss2 8214 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
9 | ssfi 9206 | . . . 4 ⊢ (({𝑋} ∈ Fin ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) supp 0 ) ∈ Fin) | |
10 | 2, 8, 9 | sylancr 585 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) supp 0 ) ∈ Fin) |
11 | funmpt 6596 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
12 | 7 | mptexd 7242 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) ∈ V) |
13 | sniffsupp.0 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑊) | |
14 | funisfsupp 9401 | . . . 4 ⊢ ((Fun (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) ∈ V ∧ 0 ∈ 𝑊) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) supp 0 ) ∈ Fin)) | |
15 | 11, 12, 13, 14 | mp3an2i 1462 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) supp 0 ) ∈ Fin)) |
16 | 10, 15 | mpbird 256 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) finSupp 0 ) |
17 | 1, 16 | eqbrtrid 5187 | 1 ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 ifcif 4532 {csn 4632 class class class wbr 5152 ↦ cmpt 5235 Fun wfun 6547 (class class class)co 7426 supp csupp 8173 Fincfn 8972 finSupp cfsupp 9395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-supp 8174 df-1o 8495 df-en 8973 df-fin 8976 df-fsupp 9396 |
This theorem is referenced by: dprdfid 19988 snifpsrbag 21869 evlsbagval 41848 mhpind 41876 cantnfresb 42802 mnringmulrcld 43714 |
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