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| Mirrors > Home > MPE Home > Th. List > suppssdm | Structured version Visualization version GIF version | ||
| Description: The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.) |
| Ref | Expression |
|---|---|
| suppssdm | ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | suppval 8187 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}}) | |
| 2 | ssrab2 4080 | . . 3 ⊢ {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}} ⊆ dom 𝐹 | |
| 3 | 1, 2 | eqsstrdi 4028 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
| 4 | supp0prc 8188 | . . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
| 5 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ dom 𝐹 | |
| 6 | 4, 5 | eqsstrdi 4028 | . 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
| 7 | 3, 6 | pm2.61i 182 | 1 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 {csn 4626 dom cdm 5685 “ cima 5688 (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-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-rab 3437 df-v 3482 df-sbc 3789 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-br 5144 df-opab 5206 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-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8186 |
| This theorem is referenced by: snopsuppss 8204 wemapso2lem 9592 cantnfcl 9707 cantnfle 9711 cantnflt 9712 cantnff 9714 cantnfres 9717 cantnfp1lem3 9720 cantnflem1b 9726 cantnflem1 9729 cantnflem3 9731 cnfcomlem 9739 cnfcom 9740 cnfcom3lem 9743 cnfcom3 9744 fsuppmapnn0fiublem 14031 fsuppmapnn0fiub 14032 gsumval3lem1 19923 gsumval3lem2 19924 gsumval3 19925 gsumzres 19927 gsumzcl2 19928 gsumzf1o 19930 gsumzaddlem 19939 gsumconst 19952 gsumzoppg 19962 gsum2d 19990 dpjidcl 20078 gsumfsum 21452 regsumsupp 21640 frlmlbs 21817 psrass1lem 21952 psrass1 21984 psrass23l 21987 psrcom 21988 psrass23 21989 mplcoe1 22055 psropprmul 22239 coe1mul2 22272 tsmsgsum 24147 rrxcph 25426 rrxsuppss 25437 rrxmval 25439 mdegfval 26101 mdegleb 26103 mdegldg 26105 deg1mul3le 26156 wilthlem3 27113 suppovss 32690 fressupp 32697 ressupprn 32699 supppreima 32700 fsupprnfi 32701 fsuppcurry1 32736 fsuppcurry2 32737 gsumfs2d 33058 gsumhashmul 33064 elrgspnlem4 33249 elrgspnsubrunlem1 33251 elrgspnsubrunlem2 33252 elrspunidl 33456 rprmdvdsprod 33562 1arithidom 33565 fedgmullem1 33680 fldextrspunlsplem 33723 fldextrspunlsp 33724 zarcmplem 33880 fdivmpt 48461 fdivmptf 48462 refdivmptf 48463 fdivpm 48464 refdivpm 48465 |
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