![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > suppss | Structured version Visualization version GIF version |
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.) |
Ref | Expression |
---|---|
suppss.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
suppss.n | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
Ref | Expression |
---|---|
suppss | ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppss.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffnd 6737 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | 2 | adantl 481 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝐹 Fn 𝐴) |
4 | simpll 767 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝐹 ∈ V) | |
5 | simplr 769 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V) | |
6 | elsuppfng 8192 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) | |
7 | 3, 4, 5, 6 | syl3anc 1370 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) |
8 | eldif 3972 | . . . . . . . . 9 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) | |
9 | suppss.n | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) | |
10 | 9 | adantll 714 | . . . . . . . . 9 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
11 | 8, 10 | sylan2br 595 | . . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
12 | 11 | expr 456 | . . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → (𝐹‘𝑘) = 𝑍)) |
13 | 12 | necon1ad 2954 | . . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) ≠ 𝑍 → 𝑘 ∈ 𝑊)) |
14 | 13 | expimpd 453 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → 𝑘 ∈ 𝑊)) |
15 | 7, 14 | sylbid 240 | . . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) → 𝑘 ∈ 𝑊)) |
16 | 15 | ssrdv 4000 | . . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ 𝑊) |
17 | 16 | ex 412 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)) |
18 | supp0prc 8186 | . . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
19 | 0ss 4405 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
20 | 18, 19 | eqsstrdi 4049 | . . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ 𝑊) |
21 | 20 | a1d 25 | . 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)) |
22 | 17, 21 | pm2.61i 182 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 ∅c0 4338 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-supp 8184 |
This theorem is referenced by: suppofssd 8226 suppcoss 8230 fsuppco2 9440 fsuppcor 9441 cantnfp1lem1 9715 cantnfp1lem3 9717 gsumzaddlem 19953 gsumzmhm 19969 gsum2d2lem 20005 lcomfsupp 20916 frlmssuvc1 21831 frlmsslsp 21833 frlmup2 21836 psrbaglesupp 21959 mvrcl 22029 mplsubglem 22036 mpllsslem 22037 mplsubrglem 22041 evlslem3 22121 mhpvscacl 22175 deg1mul3le 26170 jensen 27046 suppovss 32695 fsuppcurry1 32742 fsuppcurry2 32743 resf1o 32747 suppssnn0 32814 elrgspnlem2 33232 fedgmullem1 33656 cantnfub 43310 cantnfresb 43313 |
Copyright terms: Public domain | W3C validator |