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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 8841 | . . . 4 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | relprcnfsupp 8838 | . . . . . . . . . . . 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 514 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
8 | 7 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
9 | ressuppss 7851 | . . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
10 | ssfi 8740 | . . . . . . . . 9 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin) | |
11 | 10 | expcom 416 | . . . . . . . 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 416 | . . . . . 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 409 | . . . 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 6399 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑋)) | |
19 | 18 | adantr 483 | . . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin) → Fun (𝐹 ↾ 𝑋)) |
20 | 1, 2, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑋)) |
21 | resexg 5900 | . . . 4 ⊢ (𝐹 ∈ V → (𝐹 ↾ 𝑋) ∈ V) | |
22 | 1, 4, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) ∈ V) |
23 | funisfsupp 8840 | . . 3 ⊢ ((Fun (𝐹 ↾ 𝑋) ∧ (𝐹 ↾ 𝑋) ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) | |
24 | 20, 22, 6, 23 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
25 | 17, 24 | mpbird 259 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 ↾ cres 5559 Fun wfun 6351 (class class class)co 7158 supp csupp 7832 Fincfn 8511 finSupp cfsupp 8835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-supp 7833 df-er 8291 df-en 8512 df-fin 8515 df-fsupp 8836 |
This theorem is referenced by: dprdfadd 19144 frlmsplit2 20919 fmptssfisupp 30430 gsumle 30727 lindslinindimp2lem3 44522 lindslinindsimp2lem5 44524 |
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