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Mirrors > Home > MPE Home > Th. List > suppss | 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 AV, 28-May-2019.) |
Ref | Expression |
---|---|
suppss.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
suppss.n | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
Ref | Expression |
---|---|
suppss | ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppss.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffnd 6508 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | 2 | adantl 484 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝐹 Fn 𝐴) |
4 | fdm 6515 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | dmexg 7606 | . . . . . . . . . 10 ⊢ (𝐹 ∈ V → dom 𝐹 ∈ V) | |
6 | 5 | adantr 483 | . . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → dom 𝐹 ∈ V) |
7 | eleq1 2899 | . . . . . . . . . 10 ⊢ (𝐴 = dom 𝐹 → (𝐴 ∈ V ↔ dom 𝐹 ∈ V)) | |
8 | 7 | eqcoms 2828 | . . . . . . . . 9 ⊢ (dom 𝐹 = 𝐴 → (𝐴 ∈ V ↔ dom 𝐹 ∈ V)) |
9 | 6, 8 | syl5ibr 248 | . . . . . . . 8 ⊢ (dom 𝐹 = 𝐴 → ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝐴 ∈ V)) |
10 | 1, 4, 9 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝐴 ∈ V)) |
11 | 10 | impcom 410 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝐴 ∈ V) |
12 | simplr 767 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V) | |
13 | elsuppfn 7831 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ 𝑍 ∈ V) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) | |
14 | 3, 11, 12, 13 | syl3anc 1366 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍))) |
15 | eldif 3939 | . . . . . . . . 9 ⊢ (𝑘 ∈ (𝐴 ∖ 𝑊) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) | |
16 | suppss.n | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) | |
17 | 16 | adantll 712 | . . . . . . . . 9 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
18 | 15, 17 | sylan2br 596 | . . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝑊)) → (𝐹‘𝑘) = 𝑍) |
19 | 18 | expr 459 | . . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 ∈ 𝑊 → (𝐹‘𝑘) = 𝑍)) |
20 | 19 | necon1ad 3032 | . . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) ≠ 𝑍 → 𝑘 ∈ 𝑊)) |
21 | 20 | expimpd 456 | . . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → 𝑘 ∈ 𝑊)) |
22 | 14, 21 | sylbid 242 | . . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝑘 ∈ (𝐹 supp 𝑍) → 𝑘 ∈ 𝑊)) |
23 | 22 | ssrdv 3966 | . . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ 𝑊) |
24 | 23 | ex 415 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)) |
25 | supp0prc 7826 | . . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
26 | 0ss 4343 | . . . 4 ⊢ ∅ ⊆ 𝑊 | |
27 | 25, 26 | eqsstrdi 4014 | . . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ 𝑊) |
28 | 27 | a1d 25 | . 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)) |
29 | 24, 28 | pm2.61i 184 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 Vcvv 3491 ∖ cdif 3926 ⊆ wss 3929 ∅c0 4284 dom cdm 5548 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 supp csupp 7823 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-supp 7824 |
This theorem is referenced by: suppofssd 7860 fsuppco2 8859 fsuppcor 8860 cantnfp1lem1 9134 cantnfp1lem3 9136 gsumzaddlem 19036 gsumzmhm 19052 gsum2d2lem 19088 lcomfsupp 19669 psrbaglesupp 20143 mplsubglem 20209 mpllsslem 20210 mplsubrglem 20214 mvrcl 20224 evlslem3 20288 mhpvscacl 20336 frlmssuvc1 20933 frlmsslsp 20935 frlmup2 20938 deg1mul3le 24708 jensen 25564 suppovss 30426 fsuppcurry1 30461 fsuppcurry2 30462 resf1o 30466 fedgmullem1 31049 |
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