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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 8550 | . . . 4 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | relprcnfsupp 8547 | . . . . . . . . . . . 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 509 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
8 | 7 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
9 | ressuppss 7578 | . . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
10 | ssfi 8449 | . . . . . . . . 9 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin) | |
11 | 10 | expcom 404 | . . . . . . . 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 404 | . . . . . 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 397 | . . . 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 6165 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑋)) | |
19 | 18 | adantr 474 | . . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin) → Fun (𝐹 ↾ 𝑋)) |
20 | 1, 2, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑋)) |
21 | resexg 5679 | . . . 4 ⊢ (𝐹 ∈ V → (𝐹 ↾ 𝑋) ∈ V) | |
22 | 1, 4, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) ∈ V) |
23 | funisfsupp 8549 | . . 3 ⊢ ((Fun (𝐹 ↾ 𝑋) ∧ (𝐹 ↾ 𝑋) ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) | |
24 | 20, 22, 6, 23 | syl3anc 1496 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
25 | 17, 24 | mpbird 249 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 Vcvv 3414 ⊆ wss 3798 class class class wbr 4873 ↾ cres 5344 Fun wfun 6117 (class class class)co 6905 supp csupp 7559 Fincfn 8222 finSupp cfsupp 8544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-supp 7560 df-er 8009 df-en 8223 df-fin 8226 df-fsupp 8545 |
This theorem is referenced by: dprdfadd 18773 frlmsplit2 20479 gsumle 30324 lindslinindimp2lem3 43096 lindslinindsimp2lem5 43098 |
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