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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 3733 | . 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) | |
2 | fvex 6344 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
3 | eldifsn 4454 | . . . . . 6 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)) | |
4 | 2, 3 | mpbiran 688 | . . . . 5 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑋) ≠ 𝑍) |
5 | suppssr.f | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
6 | 5 | ffnd 6185 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | suppssr.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | suppssr.z | . . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
9 | elsuppfn 7458 | . . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
10 | 6, 7, 8, 9 | syl3anc 1476 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) |
11 | ibar 518 | . . . . . . . . . . 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 | syl6bbr 278 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))) |
14 | 13 | pm5.32da 568 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))) |
15 | 10, 14 | bitrd 268 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))) |
16 | suppssr.n | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) | |
17 | 16 | sseld 3751 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊)) |
18 | 15, 17 | sylbird 250 | . . . . . 6 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})) → 𝑋 ∈ 𝑊)) |
19 | 18 | expdimp 440 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) → 𝑋 ∈ 𝑊)) |
20 | 4, 19 | syl5bir 233 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊)) |
21 | 20 | necon1bd 2961 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍)) |
22 | 21 | impr 442 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
23 | 1, 22 | sylan2b 581 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∖ cdif 3720 ⊆ wss 3723 {csn 4317 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 supp csupp 7450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-supp 7451 |
This theorem is referenced by: fsuppmptif 8465 fsuppco2 8468 fsuppcor 8469 cantnfp1lem1 8743 cantnfp1lem3 8745 cantnflem1 8754 cnfcom2lem 8766 gsumval3 18515 gsumcllem 18516 gsumzaddlem 18528 gsumzmhm 18544 gsumpt 18568 gsum2dlem1 18576 gsum2dlem2 18577 gsum2d 18578 dprdfinv 18626 dprdfadd 18627 dmdprdsplitlem 18644 dpjidcl 18665 gsumdixp 18817 lcomfsupp 19113 psrbaglesupp 19583 psrbagaddcl 19585 psrbaglefi 19587 mplsubglem 19649 mpllsslem 19650 mplsubrglem 19654 mplmonmul 19679 mplcoe1 19680 mplcoe5 19683 mplbas2 19685 evlslem4 19723 evlslem2 19727 uvcresum 20349 frlmsslsp 20352 rrxcph 23399 rrxmval 23407 rrxmetlem 23409 rrxmet 23410 rrxdstprj1 23411 deg1mul3le 24096 eulerpartlemb 30770 |
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