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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 6329 | . . . . . . 7 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
4 | 3 | simprbi 500 | . . . . . 6 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun ◡𝐺) |
6 | cofunex2g 7633 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V) | |
7 | 1, 5, 6 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
8 | fsuppco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
9 | suppimacnv 7824 | . . . 4 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) | |
10 | 7, 8, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) |
11 | suppimacnv 7824 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
12 | 1, 8, 11 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
13 | fsuppco.f | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
14 | 13 | fsuppimpd 8824 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
15 | 12, 14 | eqeltrrd 2891 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
16 | 15, 2 | fsuppcolem 8848 | . . 3 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
17 | 10, 16 | eqeltrd 2890 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin) |
18 | fsuppimp 8823 | . . . . . 6 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
19 | 18 | simpld 498 | . . . . 5 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹) |
20 | 13, 19 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
21 | f1fun 6551 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺) | |
22 | 2, 21 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐺) |
23 | funco 6364 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
24 | 20, 22, 23 | syl2anc 587 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
25 | funisfsupp 8822 | . . 3 ⊢ ((Fun (𝐹 ∘ 𝐺) ∧ (𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) | |
26 | 24, 7, 8, 25 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) |
27 | 17, 26 | mpbird 260 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 {csn 4525 class class class wbr 5030 ◡ccnv 5518 “ cima 5522 ∘ ccom 5523 Fun wfun 6318 ⟶wf 6320 –1-1→wf1 6321 (class class class)co 7135 supp csupp 7813 Fincfn 8492 finSupp cfsupp 8817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-supp 7814 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-fin 8496 df-fsupp 8818 |
This theorem is referenced by: mapfienlem1 8852 mapfienlem2 8853 coe1sfi 20842 gsumpart 30740 |
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