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Mirrors > Home > MPE Home > Th. List > fsuppres | Structured version Visualization version GIF version |
Description: The restriction of a finitely supported function is finitely supported. (Contributed by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
fsuppres.s | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppres.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
fsuppres | ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppres.s | . . 3 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
2 | fsuppimp 8823 | . . . 4 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | relprcnfsupp 8820 | . . . . . . . . . . . 12 ⊢ (¬ 𝐹 ∈ V → ¬ 𝐹 finSupp 𝑍) | |
4 | 3 | con4i 114 | . . . . . . . . . . 11 ⊢ (𝐹 finSupp 𝑍 → 𝐹 ∈ V) |
5 | 1, 4 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | fsuppres.z | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
7 | 5, 6 | jca 515 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
8 | 7 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
9 | ressuppss 7832 | . . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
10 | ssfi 8722 | . . . . . . . . 9 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin) | |
11 | 10 | expcom 417 | . . . . . . . 8 ⊢ (((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍) → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
12 | 8, 9, 11 | 3syl 18 | . . . . . . 7 ⊢ ((𝜑 ∧ Fun 𝐹) → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
13 | 12 | expcom 417 | . . . . . 6 ⊢ (Fun 𝐹 → (𝜑 → ((𝐹 supp 𝑍) ∈ Fin → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin))) |
14 | 13 | com23 86 | . . . . 5 ⊢ (Fun 𝐹 → ((𝐹 supp 𝑍) ∈ Fin → (𝜑 → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin))) |
15 | 14 | imp 410 | . . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin) → (𝜑 → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
16 | 2, 15 | syl 17 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (𝜑 → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
17 | 1, 16 | mpcom 38 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin) |
18 | funres 6366 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑋)) | |
19 | 18 | adantr 484 | . . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin) → Fun (𝐹 ↾ 𝑋)) |
20 | 1, 2, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑋)) |
21 | resexg 5864 | . . . 4 ⊢ (𝐹 ∈ V → (𝐹 ↾ 𝑋) ∈ V) | |
22 | 1, 4, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) ∈ V) |
23 | funisfsupp 8822 | . . 3 ⊢ ((Fun (𝐹 ↾ 𝑋) ∧ (𝐹 ↾ 𝑋) ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) | |
24 | 20, 22, 6, 23 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
25 | 17, 24 | mpbird 260 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↾ cres 5521 Fun wfun 6318 (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-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-rab 3115 df-v 3443 df-sbc 3721 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-br 5031 df-opab 5093 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: dprdfadd 19135 frlmsplit2 20462 fmptssfisupp 30445 gsumle 30775 zarcmplem 31234 fsuppssind 39459 lindslinindimp2lem3 44869 lindslinindsimp2lem5 44871 |
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