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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 9401 | . . . 4 ⊢ Rel finSupp | |
2 | fsuppss.2 | . . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
3 | brrelex1 5742 | . . . 4 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝐺 ∈ V) | |
4 | 1, 2, 3 | sylancr 587 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
5 | fsuppss.1 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝐺) | |
6 | 4, 5 | ssexd 5330 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
7 | brrelex2 5743 | . . 3 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝑍 ∈ V) | |
8 | 1, 2, 7 | sylancr 587 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
9 | 2 | fsuppfund 9408 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
10 | funss 6587 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
11 | 5, 9, 10 | sylc 65 | . 2 ⊢ (𝜑 → Fun 𝐹) |
12 | funsssuppss 8214 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)) | |
13 | 9, 5, 4, 12 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)) |
14 | 6, 8, 11, 2, 13 | fsuppsssuppgd 9420 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 Rel wrel 5694 Fun wfun 6557 (class class class)co 7431 supp csupp 8184 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: mhphf 42584 |
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