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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 2737 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
| 2 | suppss2.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ 𝑉) |
| 4 | simpl 482 | . . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V) | |
| 5 | 1, 3, 4 | mptsuppdifd 8211 | . . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = {𝑘 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) |
| 6 | eldifsni 4790 | . . . . . . 7 ⊢ (𝐵 ∈ (V ∖ {𝑍}) → 𝐵 ≠ 𝑍) | |
| 7 | eldif 3961 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) | |
| 8 | suppss2.n | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) | |
| 9 | 8 | adantll 714 | . . . . . . . . . 10 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
| 10 | 7, 9 | sylan2br 595 | . . . . . . . . 9 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → 𝐵 = 𝑍) |
| 11 | 10 | expr 456 | . . . . . . . 8 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → 𝐵 = 𝑍)) |
| 12 | 11 | necon1ad 2957 | . . . . . . 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 4075 | . . . 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 412 | . 2 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)) |
| 18 | id 22 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V) | |
| 19 | 18 | intnand 488 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ 𝑍 ∈ V)) |
| 20 | supp0prc 8188 | . . . . 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 4400 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
| 23 | 21, 22 | eqsstrdi 4028 | . . 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 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 {csn 4626 ↦ cmpt 5225 (class class class)co 7431 supp csupp 8185 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8186 |
| This theorem is referenced by: suppsssn 8226 fsuppmptif 9439 sniffsupp 9440 cantnflem1d 9728 cantnflem1 9729 gsumzsplit 19945 gsummpt1n0 19983 gsum2dlem1 19988 gsum2dlem2 19989 gsum2d 19990 dprdfid 20037 dprdfinv 20039 dprdfadd 20040 dmdprdsplitlem 20057 dpjidcl 20078 uvcff 21811 uvcresum 21813 psrlidm 21982 psrridm 21983 mplsubrg 22025 mplmon 22053 mplmonmul 22054 mplcoe1 22055 mplcoe5 22058 mplbas2 22060 evlslem4 22100 evlslem2 22103 evlslem3 22104 evlslem1 22106 coe1tmmul2 22279 coe1tmmul 22280 evls1fpws 22373 tsmssplit 24160 coe1mul3 26138 plypf1 26251 tayl0 26403 suppss2f 32648 suppss3 32735 gsummptres2 33056 elrgspnlem1 33246 elrgspnlem2 33247 elrgspnlem3 33248 elrgspnsubrunlem2 33252 elrspunidl 33456 elrspunsn 33457 fedgmullem2 33681 fldextrspunlsp 33724 evlsvvvallem 42571 evlsvvvallem2 42572 evlsvvval 42573 evlsbagval 42576 selvvvval 42595 evlselv 42597 mhpind 42604 evlsmhpvvval 42605 |
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