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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > offinsupp1 | Structured version Visualization version GIF version |
Description: Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.) |
Ref | Expression |
---|---|
offinsupp1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offinsupp1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
offinsupp1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
offinsupp1.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
offinsupp1.g | ⊢ (𝜑 → 𝐺:𝐴⟶𝑇) |
offinsupp1.1 | ⊢ (𝜑 → 𝐹 finSupp 𝑌) |
offinsupp1.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍) |
Ref | Expression |
---|---|
offinsupp1 | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offinsupp1.1 | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑌) | |
2 | 1 | fsuppimpd 9319 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑌) ∈ Fin) |
3 | ssidd 3971 | . . . 4 ⊢ (𝜑 → (𝐹 supp 𝑌) ⊆ (𝐹 supp 𝑌)) | |
4 | offinsupp1.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍) | |
5 | offinsupp1.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
6 | offinsupp1.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐴⟶𝑇) | |
7 | offinsupp1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | offinsupp1.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
9 | 3, 4, 5, 6, 7, 8 | suppssof1 8134 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑌)) |
10 | 2, 9 | ssfid 9217 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin) |
11 | ovexd 7396 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑆 ∧ 𝑗 ∈ 𝑇)) → (𝑖𝑅𝑗) ∈ V) | |
12 | inidm 4182 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
13 | 11, 5, 6, 7, 7, 12 | off 7639 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐴⟶V) |
14 | 13 | ffund 6676 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺)) |
15 | ovexd 7396 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) ∈ V) | |
16 | offinsupp1.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
17 | funisfsupp 9317 | . . 3 ⊢ ((Fun (𝐹 ∘f 𝑅𝐺) ∧ (𝐹 ∘f 𝑅𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin)) | |
18 | 14, 15, 16, 17 | syl3anc 1372 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin)) |
19 | 10, 18 | mpbird 257 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 class class class wbr 5109 Fun wfun 6494 ⟶wf 6496 (class class class)co 7361 ∘f cof 7619 supp csupp 8096 Fincfn 8889 finSupp cfsupp 9311 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-supp 8097 df-1o 8416 df-en 8890 df-fin 8893 df-fsupp 9312 |
This theorem is referenced by: fedgmullem1 32388 |
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