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Mirrors > Home > MPE Home > Th. List > suppss2 | 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 Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
suppss2.n | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
suppss2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
suppss2 | ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
2 | suppss2.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 2 | adantl 483 | . . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ 𝑉) |
4 | simpl 484 | . . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V) | |
5 | 1, 3, 4 | mptsuppdifd 8171 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
6 | eldifsni 4794 | . . . . . . 7 ⊢ (𝐵 ∈ (V ∖ {𝑍}) → 𝐵 ≠ 𝑍) | |
7 | eldif 3959 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) | |
8 | suppss2.n | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) | |
9 | 8 | adantll 713 | . . . . . . . . . 10 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
10 | 7, 9 | sylan2br 596 | . . . . . . . . 9 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → 𝐵 = 𝑍) |
11 | 10 | expr 458 | . . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → 𝐵 = 𝑍)) |
12 | 11 | necon1ad 2958 | . . . . . . 7 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (𝐵 ≠ 𝑍 → 𝑘 ∈ 𝑊)) |
13 | 6, 12 | syl5 34 | . . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ (V ∖ {𝑍}) → 𝑘 ∈ 𝑊)) |
14 | 13 | 3impia 1118 | . . . . 5 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴 ∧ 𝐵 ∈ (V ∖ {𝑍})) → 𝑘 ∈ 𝑊) |
15 | 14 | rabssdv 4073 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} ⊆ 𝑊) |
16 | 5, 15 | eqsstrd 4021 | . . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
17 | 16 | ex 414 | . 2 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)) |
18 | id 22 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V) | |
19 | 18 | intnand 490 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ 𝑍 ∈ V)) |
20 | supp0prc 8149 | . . . . 5 ⊢ (¬ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ 𝑍 ∈ V) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = ∅) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = ∅) |
22 | 0ss 4397 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
23 | 21, 22 | eqsstrdi 4037 | . . 3 ⊢ (¬ 𝑍 ∈ V → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
24 | 23 | a1d 25 | . 2 ⊢ (¬ 𝑍 ∈ V → (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)) |
25 | 17, 24 | pm2.61i 182 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 Vcvv 3475 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 {csn 4629 ↦ cmpt 5232 (class class class)co 7409 supp csupp 8146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 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-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 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-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-supp 8147 |
This theorem is referenced by: suppsssn 8186 fsuppmptif 9394 sniffsupp 9395 cantnflem1d 9683 cantnflem1 9684 gsumzsplit 19795 gsummpt1n0 19833 gsum2dlem1 19838 gsum2dlem2 19839 gsum2d 19840 dprdfid 19887 dprdfinv 19889 dprdfadd 19890 dmdprdsplitlem 19907 dpjidcl 19928 uvcff 21346 uvcresum 21348 psrbagaddclOLD 21482 psrlidm 21523 psrridm 21524 mplsubrg 21564 mplmon 21590 mplmonmul 21591 mplcoe1 21592 mplcoe5 21595 mplbas2 21597 evlslem4 21637 evlslem2 21642 evlslem3 21643 evlslem1 21645 coe1tmmul2 21798 coe1tmmul 21799 tsmssplit 23656 coe1mul3 25617 plypf1 25726 tayl0 25874 suppss2f 31863 suppss3 31949 gsummptres2 32205 elrspunidl 32546 elrspunsn 32547 evls1fpws 32646 fedgmullem2 32715 evlsvvvallem 41133 evlsvvvallem2 41134 evlsvvval 41135 evlsbagval 41138 selvvvval 41157 evlselv 41159 mhpind 41166 evlsmhpvvval 41167 |
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