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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 7905 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}}) | |
2 | ssrab2 3993 | . . 3 ⊢ {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}} ⊆ dom 𝐹 | |
3 | 1, 2 | eqsstrdi 3955 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
4 | supp0prc 7906 | . . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
5 | 0ss 4311 | . . 3 ⊢ ∅ ⊆ dom 𝐹 | |
6 | 4, 5 | eqsstrdi 3955 | . 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
7 | 3, 6 | pm2.61i 185 | 1 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∈ wcel 2110 ≠ wne 2940 {crab 3065 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 {csn 4541 dom cdm 5551 “ cima 5554 (class class class)co 7213 supp csupp 7903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-supp 7904 |
This theorem is referenced by: snopsuppss 7921 wemapso2lem 9168 cantnfcl 9282 cantnfle 9286 cantnflt 9287 cantnff 9289 cantnfres 9292 cantnfp1lem3 9295 cantnflem1b 9301 cantnflem1 9304 cantnflem3 9306 cnfcomlem 9314 cnfcom 9315 cnfcom3lem 9318 cnfcom3 9319 fsuppmapnn0fiublem 13563 fsuppmapnn0fiub 13564 gsumval3lem1 19290 gsumval3lem2 19291 gsumval3 19292 gsumzres 19294 gsumzcl2 19295 gsumzf1o 19297 gsumzaddlem 19306 gsumconst 19319 gsumzoppg 19329 gsum2d 19357 dpjidcl 19445 gsumfsum 20430 regsumsupp 20584 frlmlbs 20759 psrass1lemOLD 20899 psrass1lem 20902 psrass1 20930 psrass23l 20933 psrcom 20934 psrass23 20935 mplcoe1 20994 psropprmul 21159 coe1mul2 21190 tsmsgsum 23036 rrxcph 24289 rrxsuppss 24300 rrxmval 24302 mdegfval 24960 mdegleb 24962 mdegldg 24964 deg1mul3le 25014 wilthlem3 25952 suppovss 30737 fressupp 30742 ressupprn 30744 supppreima 30745 fsupprnfi 30746 fsuppcurry1 30780 fsuppcurry2 30781 gsumhashmul 31035 elrspunidl 31320 fedgmullem1 31424 zarcmplem 31545 fdivmpt 45559 fdivmptf 45560 refdivmptf 45561 fdivpm 45562 refdivpm 45563 |
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