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Mirrors > Home > MPE Home > Th. List > suppssr | Structured version Visualization version GIF version |
Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
suppssr.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
suppssr.n | ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
suppssr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
suppssr.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
Ref | Expression |
---|---|
suppssr | ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3972 | . 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) | |
2 | fvex 6919 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
3 | eldifsn 4790 | . . . . . 6 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)) | |
4 | 2, 3 | mpbiran 709 | . . . . 5 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑋) ≠ 𝑍) |
5 | suppssr.f | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
6 | 5 | ffnd 6737 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | suppssr.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | suppssr.z | . . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
9 | elsuppfn 8193 | . . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
10 | 6, 7, 8, 9 | syl3anc 1370 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) |
11 | ibar 528 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑋) ∈ V → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
12 | 2, 11 | mp1i 13 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))) |
13 | 12, 3 | bitr4di 289 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))) |
14 | 13 | pm5.32da 579 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))) |
15 | 10, 14 | bitrd 279 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))) |
16 | suppssr.n | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) | |
17 | 16 | sseld 3993 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊)) |
18 | 15, 17 | sylbird 260 | . . . . . 6 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})) → 𝑋 ∈ 𝑊)) |
19 | 18 | expdimp 452 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) → 𝑋 ∈ 𝑊)) |
20 | 4, 19 | biimtrrid 243 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊)) |
21 | 20 | necon1bd 2955 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍)) |
22 | 21 | impr 454 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
23 | 1, 22 | sylan2b 594 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-supp 8184 |
This theorem is referenced by: fsuppmptif 9436 fsuppco2 9440 fsuppcor 9441 cantnfp1lem1 9715 cantnfp1lem3 9717 cantnflem1 9726 cnfcom2lem 9738 gsumval3 19939 gsumcllem 19940 gsumzaddlem 19953 gsumzmhm 19969 gsumpt 19994 gsum2dlem1 20002 gsum2dlem2 20003 gsum2d 20004 gsumxp2 20012 dprdfinv 20053 dprdfadd 20054 dmdprdsplitlem 20071 dpjidcl 20092 gsumdixp 20332 lcomfsupp 20916 uvcresum 21830 frlmsslsp 21833 mplsubglem 22036 mpllsslem 22037 mplsubrglem 22041 mplmonmul 22071 mplcoe1 22072 mplcoe5 22075 mplbas2 22077 evlslem4 22117 evlslem2 22120 rrxcph 25439 rrxmval 25452 rrxmetlem 25454 rrxmet 25455 rrxdstprj1 25456 deg1mul3le 26170 suppovss 32695 elrspunidl 33435 fedgmullem1 33656 eulerpartlemb 34349 evlsvvvallem 42547 evlsvvval 42549 evlselv 42573 fsuppssindlem1 42577 evlsmhpvvval 42581 |
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