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Mirrors > Home > MPE Home > Th. List > fsuppco | Structured version Visualization version GIF version |
Description: The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.) |
Ref | Expression |
---|---|
fsuppco.f | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppco.g | ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) |
fsuppco.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppco.v | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
fsuppco | ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppco.v | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | fsuppco.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
3 | df-f1 6506 | . . . . . . 7 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
4 | 3 | simprbi 498 | . . . . . 6 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun ◡𝐺) |
6 | cofunex2g 7887 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V) | |
7 | 1, 5, 6 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
8 | fsuppco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
9 | suppimacnv 8110 | . . . 4 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) | |
10 | 7, 8, 9 | syl2anc 585 | . . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) |
11 | suppimacnv 8110 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
12 | 1, 8, 11 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
13 | fsuppco.f | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
14 | 13 | fsuppimpd 9319 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
15 | 12, 14 | eqeltrrd 2839 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
16 | 15, 2 | fsuppcolem 9344 | . . 3 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
17 | 10, 16 | eqeltrd 2838 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin) |
18 | fsuppimp 9318 | . . . . . 6 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
19 | 18 | simpld 496 | . . . . 5 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹) |
20 | 13, 19 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
21 | f1fun 6745 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺) | |
22 | 2, 21 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐺) |
23 | funco 6546 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
24 | 20, 22, 23 | syl2anc 585 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
25 | funisfsupp 9317 | . . 3 ⊢ ((Fun (𝐹 ∘ 𝐺) ∧ (𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) | |
26 | 24, 7, 8, 25 | syl3anc 1372 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) |
27 | 17, 26 | mpbird 257 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∖ cdif 3912 {csn 4591 class class class wbr 5110 ◡ccnv 5637 “ cima 5641 ∘ ccom 5642 Fun wfun 6495 ⟶wf 6497 –1-1→wf1 6498 (class class class)co 7362 supp csupp 8097 Fincfn 8890 finSupp cfsupp 9312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-supp 8098 df-1o 8417 df-en 8891 df-fin 8894 df-fsupp 9313 |
This theorem is referenced by: mapfienlem1 9348 mapfienlem2 9349 coe1sfi 21600 gsumpart 31939 |
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