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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 6585 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
4 | resindm 6050 | . . 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 6599 | . . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
7 | fnresin2 6695 | . . . 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 9300 | . . . 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 9416 | . 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍) |
14 | 5, 13 | eqbrtrrd 5172 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 Rel wrel 5694 Fun wfun 6557 Fn wfn 6558 Fincfn 8984 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: xrge0tsmsd 33048 ply1degltdimlem 33650 |
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