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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 7184 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}}) | |
2 | ssrab2 3650 | . . 3 ⊢ {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}} ⊆ dom 𝐹 | |
3 | 1, 2 | syl6eqss 3618 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
4 | supp0prc 7185 | . . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
5 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ dom 𝐹 | |
6 | 4, 5 | syl6eqss 3618 | . 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
7 | 3, 6 | pm2.61i 175 | 1 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {csn 4125 dom cdm 5038 “ cima 5041 (class class class)co 6549 supp csupp 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-supp 7183 |
This theorem is referenced by: snopsuppss 7197 wemapso2lem 8340 cantnfcl 8447 cantnfle 8451 cantnflt 8452 cantnff 8454 cantnfres 8457 cantnfp1lem2 8459 cantnfp1lem3 8460 cantnflem1b 8466 cantnflem1d 8468 cantnflem1 8469 cantnflem3 8471 cnfcomlem 8479 cnfcom 8480 cnfcom2lem 8481 cnfcom3lem 8483 cnfcom3 8484 fsuppmapnn0fiublem 12651 fsuppmapnn0fiub 12652 fsuppmapnn0fiubOLD 12653 gsumval3lem1 18129 gsumval3lem2 18130 gsumval3 18131 gsumzres 18133 gsumzcl2 18134 gsumzf1o 18136 gsumzaddlem 18144 gsumconst 18157 gsumzoppg 18167 gsum2d 18194 dpjidcl 18280 psrass1lem 19198 psrass1 19226 psrass23l 19229 psrcom 19230 psrass23 19231 mplcoe1 19286 psropprmul 19429 coe1mul2 19460 gsumfsum 19632 regsumsupp 19787 frlmlbs 19955 tsmsgsum 21752 rrxcph 22988 rrxsuppss 22994 rrxmval 22996 mdegfval 23626 mdegleb 23628 mdegldg 23630 deg1mul3le 23680 wilthlem3 24596 fdivmpt 42132 fdivmptf 42133 refdivmptf 42134 fdivpm 42135 refdivpm 42136 |
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