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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 6524 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | 2 | adantl 485 | . . . . . 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 7890 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) | |
7 | 3, 4, 5, 6 | syl3anc 1373 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) |
8 | eldif 3863 | . . . . . . . . 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 598 | . . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
12 | 11 | expr 460 | . . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → (𝐹‘𝑘) = 𝑍)) |
13 | 12 | necon1ad 2949 | . . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) ≠ 𝑍 → 𝑘 ∈ 𝑊)) |
14 | 13 | expimpd 457 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → 𝑘 ∈ 𝑊)) |
15 | 7, 14 | sylbid 243 | . . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) → 𝑘 ∈ 𝑊)) |
16 | 15 | ssrdv 3893 | . . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ 𝑊) |
17 | 16 | ex 416 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)) |
18 | supp0prc 7884 | . . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
19 | 0ss 4297 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
20 | 18, 19 | eqsstrdi 3941 | . . 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 185 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∖ cdif 3850 ⊆ wss 3853 ∅c0 4223 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 supp csupp 7881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-supp 7882 |
This theorem is referenced by: suppofssd 7923 suppcoss 7927 fsuppco2 8997 fsuppcor 8998 cantnfp1lem1 9271 cantnfp1lem3 9273 gsumzaddlem 19260 gsumzmhm 19276 gsum2d2lem 19312 lcomfsupp 19893 frlmssuvc1 20710 frlmsslsp 20712 frlmup2 20715 psrbaglesupp 20837 psrbaglesuppOLD 20838 mplsubglem 20915 mpllsslem 20916 mplsubrglem 20920 mvrcl 20931 evlslem3 20994 mhpvscacl 21048 deg1mul3le 24968 jensen 25825 suppovss 30691 fsuppcurry1 30734 fsuppcurry2 30735 resf1o 30739 fedgmullem1 31378 |
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