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Mirrors > Home > MPE Home > Th. List > resfifsupp | Structured version Visualization version GIF version |
Description: The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.) |
Ref | Expression |
---|---|
resfifsupp.f | ⊢ (𝜑 → Fun 𝐹) |
resfifsupp.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
resfifsupp.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
resfifsupp | ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resfifsupp.f | . . . 4 ⊢ (𝜑 → Fun 𝐹) | |
2 | funrel 6154 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
4 | resindm 5696 | . . 3 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹 ↾ 𝑋)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) = (𝐹 ↾ 𝑋)) |
6 | 1 | funfnd 6168 | . . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
7 | fnresin2 6254 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) Fn (𝑋 ∩ dom 𝐹)) |
9 | resfifsupp.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
10 | infi 8474 | . . . 4 ⊢ (𝑋 ∈ Fin → (𝑋 ∩ dom 𝐹) ∈ Fin) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝑋 ∩ dom 𝐹) ∈ Fin) |
12 | resfifsupp.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
13 | 8, 11, 12 | fndmfifsupp 8578 | . 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍) |
14 | 5, 13 | eqbrtrrd 4912 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 class class class wbr 4888 dom cdm 5357 ↾ cres 5359 Rel wrel 5362 Fun wfun 6131 Fn wfn 6132 Fincfn 8243 finSupp cfsupp 8565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-supp 7579 df-er 8028 df-en 8244 df-fin 8247 df-fsupp 8566 |
This theorem is referenced by: xrge0tsmsd 30351 |
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