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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsuppsssuppgd | Structured version Visualization version GIF version |
Description: If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9385. (Contributed by SN, 6-Mar-2025.) |
Ref | Expression |
---|---|
fsuppsssuppgd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
fsuppsssuppgd.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppsssuppgd.1 | ⊢ (𝜑 → Fun 𝐺) |
fsuppsssuppgd.2 | ⊢ (𝜑 → 𝐹 finSupp 𝑂) |
fsuppsssuppgd.3 | ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂)) |
Ref | Expression |
---|---|
fsuppsssuppgd | ⊢ (𝜑 → 𝐺 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppsssuppgd.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
2 | fsuppsssuppgd.1 | . 2 ⊢ (𝜑 → Fun 𝐺) | |
3 | fsuppsssuppgd.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
4 | fsuppsssuppgd.2 | . . 3 ⊢ (𝜑 → 𝐹 finSupp 𝑂) | |
5 | 4 | fsuppimpd 9375 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑂) ∈ Fin) |
6 | fsuppsssuppgd.3 | . 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂)) | |
7 | suppssfifsupp 9384 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ ((𝐹 supp 𝑂) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))) → 𝐺 finSupp 𝑍) | |
8 | 1, 2, 3, 5, 6, 7 | syl32anc 1377 | 1 ⊢ (𝜑 → 𝐺 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3948 class class class wbr 5148 Fun wfun 6537 (class class class)co 7412 supp csupp 8151 Fincfn 8945 finSupp cfsupp 9367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-1o 8472 df-en 8946 df-fin 8949 df-fsupp 9368 |
This theorem is referenced by: fsuppss 41535 evlsvvvallem 41599 evlsvvvallem2 41600 evlsvvval 41601 selvvvval 41623 evlselv 41625 mhphf 41635 |
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