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 8833 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑌) ∈ Fin) |
3 | ssidd 3983 | . . . 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 7856 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑌)) |
10 | 2, 9 | ssfid 8734 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin) |
11 | ovexd 7184 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑆 ∧ 𝑗 ∈ 𝑇)) → (𝑖𝑅𝑗) ∈ V) | |
12 | inidm 4188 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
13 | 11, 5, 6, 7, 7, 12 | off 7417 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐴⟶V) |
14 | 13 | ffund 6511 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺)) |
15 | ovexd 7184 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) ∈ V) | |
16 | offinsupp1.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
17 | funisfsupp 8831 | . . 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 1366 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin)) |
19 | 10, 18 | mpbird 259 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 class class class wbr 5059 Fun wfun 6342 ⟶wf 6344 (class class class)co 7149 ∘f cof 7400 supp csupp 7823 Fincfn 8502 finSupp cfsupp 8826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-supp 7824 df-er 8282 df-en 8503 df-fin 8506 df-fsupp 8827 |
This theorem is referenced by: fedgmullem1 31047 |
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