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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 9406 | . . . 4 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | relprcnfsupp 9402 | . . . . . . . . . . . 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 8207 | . . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝑋) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
10 | ssfi 9212 | . . . . . . . . 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 6610 | . . . . 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 6047 | . . . 4 ⊢ (𝐹 ∈ V → (𝐹 ↾ 𝑋) ∈ V) | |
22 | 1, 4, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) ∈ V) |
23 | funisfsupp 9405 | . . 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 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↾ cres 5691 Fun wfun 6557 (class class class)co 7431 supp csupp 8184 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-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-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-br 5149 df-opab 5211 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: fmptssfisupp 9432 dprdfadd 20055 frlmsplit2 21811 gsumle 33084 elrspunsn 33437 rprmdvdsprod 33542 zarcmplem 33842 psrbagres 42533 evlselv 42574 fsuppssind 42580 cantnf2 43315 lindslinindimp2lem3 48306 lindslinindsimp2lem5 48308 |
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