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Mirrors > Home > MPE Home > Th. List > fsuppss | Structured version Visualization version GIF version |
Description: A subset of a finitely supported function is a finitely supported function. (Contributed by SN, 8-Mar-2025.) |
Ref | Expression |
---|---|
fsuppss.1 | ⊢ (𝜑 → 𝐹 ⊆ 𝐺) |
fsuppss.2 | ⊢ (𝜑 → 𝐺 finSupp 𝑍) |
Ref | Expression |
---|---|
fsuppss | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfsupp 9393 | . . . 4 ⊢ Rel finSupp | |
2 | fsuppss.2 | . . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
3 | brrelex1 5733 | . . . 4 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝐺 ∈ V) | |
4 | 1, 2, 3 | sylancr 585 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
5 | fsuppss.1 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝐺) | |
6 | 4, 5 | ssexd 5326 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
7 | brrelex2 5734 | . . 3 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝑍 ∈ V) | |
8 | 1, 2, 7 | sylancr 585 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
9 | 2 | fsuppfund 9400 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
10 | funss 6575 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
11 | 5, 9, 10 | sylc 65 | . 2 ⊢ (𝜑 → Fun 𝐹) |
12 | funsssuppss 8199 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)) | |
13 | 9, 5, 4, 12 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)) |
14 | 6, 8, 11, 2, 13 | fsuppsssuppgd 9411 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 class class class wbr 5150 Rel wrel 5685 Fun wfun 6545 (class class class)co 7424 supp csupp 8169 finSupp cfsupp 9391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-supp 8170 df-1o 8491 df-en 8969 df-fin 8972 df-fsupp 9392 |
This theorem is referenced by: mhphf 41833 |
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