![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9371 | . . . 4 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | relprcnfsupp 9367 | . . . . . . . . . . . 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 511 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
8 | 7 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉)) |
9 | ressuppss 8171 | . . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
10 | ssfi 9176 | . . . . . . . . 9 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin) | |
11 | 10 | expcom 413 | . . . . . . . 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 413 | . . . . . 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 406 | . . . 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 6590 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑋)) | |
19 | 18 | adantr 480 | . . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin) → Fun (𝐹 ↾ 𝑋)) |
20 | 1, 2, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑋)) |
21 | resexg 6027 | . . . 4 ⊢ (𝐹 ∈ V → (𝐹 ↾ 𝑋) ∈ V) | |
22 | 1, 4, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) ∈ V) |
23 | funisfsupp 9370 | . . 3 ⊢ ((Fun (𝐹 ↾ 𝑋) ∧ (𝐹 ↾ 𝑋) ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) | |
24 | 20, 22, 6, 23 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝑋) finSupp 𝑍 ↔ ((𝐹 ↾ 𝑋) supp 𝑍) ∈ Fin)) |
25 | 17, 24 | mpbird 257 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ↾ cres 5678 Fun wfun 6537 (class class class)co 7412 supp csupp 8149 Fincfn 8942 finSupp cfsupp 9364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-supp 8150 df-1o 8469 df-en 8943 df-fin 8946 df-fsupp 9365 |
This theorem is referenced by: fmptssfisupp 9392 dprdfadd 19932 frlmsplit2 21548 gsumle 32513 elrspunsn 32822 zarcmplem 33160 psrbagres 41418 evlselv 41462 fsuppssind 41468 cantnf2 42378 lindslinindimp2lem3 47229 lindslinindsimp2lem5 47231 |
Copyright terms: Public domain | W3C validator |