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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 9358 | . . . 4 ⊢ Rel finSupp | |
2 | fsuppss.2 | . . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑍) | |
3 | brrelex1 5719 | . . . 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 5314 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
7 | brrelex2 5720 | . . 3 ⊢ ((Rel finSupp ∧ 𝐺 finSupp 𝑍) → 𝑍 ∈ V) | |
8 | 1, 2, 7 | sylancr 586 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
9 | 2 | fsuppfund 41522 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
10 | funss 6557 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
11 | 5, 9, 10 | sylc 65 | . 2 ⊢ (𝜑 → Fun 𝐹) |
12 | funsssuppss 8169 | . . 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 41523 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3940 class class class wbr 5138 Rel wrel 5671 Fun wfun 6527 (class class class)co 7401 supp csupp 8140 finSupp cfsupp 9356 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-supp 8141 df-1o 8461 df-en 8935 df-fin 8938 df-fsupp 9357 |
This theorem is referenced by: mhphf 41624 |
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