![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6565 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
4 | resindm 6029 | . . 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 6579 | . . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
7 | fnresin2 6676 | . . . 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 9291 | . . . 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 9401 | . 2 ⊢ (𝜑 → (𝐹 ↾ (𝑋 ∩ dom 𝐹)) finSupp 𝑍) |
14 | 5, 13 | eqbrtrrd 5167 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3938 class class class wbr 5143 dom cdm 5672 ↾ cres 5674 Rel wrel 5677 Fun wfun 6537 Fn wfn 6538 Fincfn 8962 finSupp cfsupp 9385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-supp 8164 df-1o 8485 df-en 8963 df-fin 8966 df-fsupp 9386 |
This theorem is referenced by: xrge0tsmsd 32816 ply1degltdimlem 33377 |
Copyright terms: Public domain | W3C validator |