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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 9362 | . . . 4 ⊢ Rel finSupp | |
2 | fsuppss.2 | . . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
3 | brrelex1 5722 | . . . 4 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝐺 ∈ V) | |
4 | 1, 2, 3 | sylancr 586 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
5 | fsuppss.1 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝐺) | |
6 | 4, 5 | ssexd 5317 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
7 | brrelex2 5723 | . . 3 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝑍 ∈ V) | |
8 | 1, 2, 7 | sylancr 586 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
9 | 2 | fsuppfund 41605 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
10 | funss 6560 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
11 | 5, 9, 10 | sylc 65 | . 2 ⊢ (𝜑 → Fun 𝐹) |
12 | funsssuppss 8172 | . . 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 41606 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 class class class wbr 5141 Rel wrel 5674 Fun wfun 6530 (class class class)co 7404 supp csupp 8143 finSupp cfsupp 9360 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-supp 8144 df-1o 8464 df-en 8939 df-fin 8942 df-fsupp 9361 |
This theorem is referenced by: mhphf 41707 |
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