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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 2798 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
2 | suppss2.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 2 | adantl 485 | . . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ 𝑉) |
4 | simpl 486 | . . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V) | |
5 | 1, 3, 4 | mptsuppdifd 7835 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
6 | eldifsni 4683 | . . . . . . 7 ⊢ (𝐵 ∈ (V ∖ {𝑍}) → 𝐵 ≠ 𝑍) | |
7 | eldif 3891 | . . . . . . . . . 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 597 | . . . . . . . . 9 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → 𝐵 = 𝑍) |
11 | 10 | expr 460 | . . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → 𝐵 = 𝑍)) |
12 | 11 | necon1ad 3004 | . . . . . . 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 1114 | . . . . 5 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴 ∧ 𝐵 ∈ (V ∖ {𝑍})) → 𝑘 ∈ 𝑊) |
15 | 14 | rabssdv 4002 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})} ⊆ 𝑊) |
16 | 5, 15 | eqsstrd 3953 | . . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
17 | 16 | ex 416 | . 2 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)) |
18 | id 22 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V) | |
19 | 18 | intnand 492 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ 𝑍 ∈ V)) |
20 | supp0prc 7816 | . . . . 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 4304 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
23 | 21, 22 | eqsstrdi 3969 | . . 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 185 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 {csn 4525 ↦ cmpt 5110 (class class class)co 7135 supp csupp 7813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-supp 7814 |
This theorem is referenced by: suppsssn 7848 fsuppmptif 8847 sniffsupp 8857 cantnflem1d 9135 cantnflem1 9136 gsumzsplit 19040 gsummpt1n0 19078 gsum2dlem1 19083 gsum2dlem2 19084 gsum2d 19085 dprdfid 19132 dprdfinv 19134 dprdfadd 19135 dmdprdsplitlem 19152 dpjidcl 19173 uvcff 20480 uvcresum 20482 psrbagaddcl 20608 psrlidm 20641 psrridm 20642 mplsubrg 20678 mplmon 20703 mplmonmul 20704 mplcoe1 20705 mplcoe5 20708 mplbas2 20710 evlslem4 20747 evlslem2 20751 evlslem3 20752 evlslem1 20754 coe1tmmul2 20905 coe1tmmul 20906 tsmssplit 22757 coe1mul3 24700 plypf1 24809 tayl0 24957 suppss2f 30398 suppss3 30486 gsummptres2 30738 elrspunidl 31014 fedgmullem2 31114 |
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