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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 8165 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
6 | eldifsni 4791 | . . . . . . 7 ⊢ (𝐵 ∈ (V ∖ {𝑍}) → 𝐵 ≠ 𝑍) | |
7 | eldif 3956 | . . . . . . . . . 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 4070 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} ⊆ 𝑊) |
16 | 5, 15 | eqsstrd 4018 | . . 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 8143 | . . . . 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 4394 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
23 | 21, 22 | eqsstrdi 4034 | . . 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 3943 ⊆ wss 3946 ∅c0 4320 {csn 4626 ↦ cmpt 5229 (class class class)co 7403 supp csupp 8140 |
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 5283 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-supp 8141 |
This theorem is referenced by: suppsssn 8180 fsuppmptif 9389 sniffsupp 9390 cantnflem1d 9678 cantnflem1 9679 gsumzsplit 19786 gsummpt1n0 19824 gsum2dlem1 19829 gsum2dlem2 19830 gsum2d 19831 dprdfid 19878 dprdfinv 19880 dprdfadd 19881 dmdprdsplitlem 19898 dpjidcl 19919 uvcff 21329 uvcresum 21331 psrbagaddclOLD 21463 psrlidm 21504 psrridm 21505 mplsubrg 21545 mplmon 21571 mplmonmul 21572 mplcoe1 21573 mplcoe5 21576 mplbas2 21578 evlslem4 21618 evlslem2 21623 evlslem3 21624 evlslem1 21626 coe1tmmul2 21779 coe1tmmul 21780 tsmssplit 23637 coe1mul3 25598 plypf1 25707 tayl0 25855 suppss2f 31840 suppss3 31926 gsummptres2 32182 elrspunidl 32503 elrspunsn 32504 evls1fpws 32595 fedgmullem2 32659 evlsvvvallem 41082 evlsvvvallem2 41083 evlsvvval 41084 evlsbagval 41087 selvvvval 41106 evlselv 41108 mhpind 41115 evlsmhpvvval 41116 |
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