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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 6919 | . . . . 5 ⊢ (𝐹‘𝑘) ∈ V | |
| 2 | fsuppmptif.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
| 3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝑊) | 
| 4 | ifexg 4575 | . . . . 5 ⊢ (((𝐹‘𝑘) ∈ V ∧ 𝑍 ∈ 𝑊) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) | |
| 5 | 1, 3, 4 | sylancr 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) | 
| 6 | 5 | fmpttd 7135 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)):𝐴⟶V) | 
| 7 | 6 | ffund 6740 | . 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍))) | 
| 8 | fsuppmptif.s | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
| 9 | 8 | fsuppimpd 9409 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) | 
| 10 | fsuppmptif.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 11 | ssidd 4007 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
| 12 | fsuppmptif.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 13 | 10, 11, 12, 2 | suppssr 8220 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑘) = 𝑍) | 
| 14 | 13 | ifeq1d 4545 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = if(𝑘 ∈ 𝐷, 𝑍, 𝑍)) | 
| 15 | ifid 4566 | . . . . 5 ⊢ if(𝑘 ∈ 𝐷, 𝑍, 𝑍) = 𝑍 | |
| 16 | 14, 15 | eqtrdi 2793 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = 𝑍) | 
| 17 | 16, 12 | suppss2 8225 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | 
| 18 | 9, 17 | ssfid 9301 | . 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin) | 
| 19 | 12 | mptexd 7244 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V) | 
| 20 | isfsupp 9405 | . . 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 584 | . 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 713 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 | 
| This theorem is referenced by: cantnflem1d 9728 gsumzsplit 19945 | 
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