|   | 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 6583 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) | 
| 4 | resindm 6048 | . . 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 6597 | . . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) | 
| 7 | fnresin2 6694 | . . . 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 9302 | . . . 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 9418 | . 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 1540 ∈ wcel 2108 ∩ cin 3950 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 Fun wfun 6555 Fn wfn 6556 Fincfn 8985 finSupp cfsupp 9401 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 | 
| This theorem is referenced by: xrge0tsmsd 33065 ply1degltdimlem 33673 | 
| Copyright terms: Public domain | W3C validator |