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Mirrors > Home > MPE Home > Th. List > fsuppmptif | Structured version Visualization version GIF version |
Description: A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
fsuppmptif.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fsuppmptif.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fsuppmptif.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppmptif.s | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Ref | Expression |
---|---|
fsuppmptif | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . . . 5 ⊢ (𝐹‘𝑘) ∈ V | |
2 | fsuppmptif.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
3 | 2 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝑊) |
4 | ifexg 4472 | . . . . 5 ⊢ (((𝐹‘𝑘) ∈ V ∧ 𝑍 ∈ 𝑊) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) | |
5 | 1, 3, 4 | sylancr 590 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) |
6 | 5 | fmpttd 6856 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)):𝐴⟶V) |
7 | 6 | ffund 6491 | . 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍))) |
8 | fsuppmptif.s | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
9 | 8 | fsuppimpd 8824 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
10 | fsuppmptif.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
11 | ssidd 3938 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
12 | fsuppmptif.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | 10, 11, 12, 2 | suppssr 7844 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑘) = 𝑍) |
14 | 13 | ifeq1d 4443 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = if(𝑘 ∈ 𝐷, 𝑍, 𝑍)) |
15 | ifid 4464 | . . . . 5 ⊢ if(𝑘 ∈ 𝐷, 𝑍, 𝑍) = 𝑍 | |
16 | 14, 15 | eqtrdi 2849 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = 𝑍) |
17 | 16, 12 | suppss2 7847 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
18 | 9, 17 | ssfid 8725 | . 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin) |
19 | 12 | mptexd 6964 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V) |
20 | isfsupp 8821 | . . 3 ⊢ (((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍 ↔ (Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin))) | |
21 | 19, 2, 20 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍 ↔ (Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin))) |
22 | 7, 18, 21 | mpbir2and 712 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supp csupp 7813 Fincfn 8492 finSupp cfsupp 8817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-supp 7814 df-er 8272 df-en 8493 df-fin 8496 df-fsupp 8818 |
This theorem is referenced by: cantnflem1d 9135 gsumzsplit 19040 |
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