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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 6719 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | 2 | adantl 480 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝐹 Fn 𝐴) |
4 | simpll 763 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝐹 ∈ V) | |
5 | simplr 765 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V) | |
6 | elsuppfng 8159 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) | |
7 | 3, 4, 5, 6 | syl3anc 1369 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) |
8 | eldif 3959 | . . . . . . . . 9 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) | |
9 | suppss.n | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) | |
10 | 9 | adantll 710 | . . . . . . . . 9 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
11 | 8, 10 | sylan2br 593 | . . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
12 | 11 | expr 455 | . . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → (𝐹‘𝑘) = 𝑍)) |
13 | 12 | necon1ad 2955 | . . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) ≠ 𝑍 → 𝑘 ∈ 𝑊)) |
14 | 13 | expimpd 452 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → 𝑘 ∈ 𝑊)) |
15 | 7, 14 | sylbid 239 | . . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) → 𝑘 ∈ 𝑊)) |
16 | 15 | ssrdv 3989 | . . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ 𝑊) |
17 | 16 | ex 411 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)) |
18 | supp0prc 8153 | . . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
19 | 0ss 4397 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
20 | 18, 19 | eqsstrdi 4037 | . . 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 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 Vcvv 3472 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 supp csupp 8150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-supp 8151 |
This theorem is referenced by: suppofssd 8192 suppcoss 8196 fsuppco2 9402 fsuppcor 9403 cantnfp1lem1 9677 cantnfp1lem3 9679 gsumzaddlem 19832 gsumzmhm 19848 gsum2d2lem 19884 lcomfsupp 20658 frlmssuvc1 21570 frlmsslsp 21572 frlmup2 21575 psrbaglesupp 21698 psrbaglesuppOLD 21699 mvrcl 21772 mplsubglem 21779 mpllsslem 21780 mplsubrglem 21784 evlslem3 21864 mhpvscacl 21918 deg1mul3le 25868 jensen 26727 suppovss 32171 fsuppcurry1 32215 fsuppcurry2 32216 resf1o 32220 suppssnn0 32282 fedgmullem1 33000 cantnfub 42375 cantnfresb 42378 |
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