Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7821 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}}) | |
2 | ssrab2 4053 | . . 3 ⊢ {𝑖 ∈ dom 𝐹 ∣ (𝐹 “ {𝑖}) ≠ {𝑍}} ⊆ dom 𝐹 | |
3 | 1, 2 | eqsstrdi 4018 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
4 | supp0prc 7822 | . . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅) | |
5 | 0ss 4347 | . . 3 ⊢ ∅ ⊆ dom 𝐹 | |
6 | 4, 5 | eqsstrdi 4018 | . 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ dom 𝐹) |
7 | 3, 6 | pm2.61i 183 | 1 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 {crab 3139 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 {csn 4557 dom cdm 5548 “ cima 5551 (class class class)co 7145 supp csupp 7819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 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-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-supp 7820 |
This theorem is referenced by: snopsuppss 7834 wemapso2lem 9004 cantnfcl 9118 cantnfle 9122 cantnflt 9123 cantnff 9125 cantnfres 9128 cantnfp1lem3 9131 cantnflem1b 9137 cantnflem1 9140 cantnflem3 9142 cnfcomlem 9150 cnfcom 9151 cnfcom3lem 9154 cnfcom3 9155 fsuppmapnn0fiublem 13346 fsuppmapnn0fiub 13347 gsumval3lem1 18954 gsumval3lem2 18955 gsumval3 18956 gsumzres 18958 gsumzcl2 18959 gsumzf1o 18961 gsumzaddlem 18970 gsumconst 18983 gsumzoppg 18993 gsum2d 19021 dpjidcl 19109 psrass1lem 20085 psrass1 20113 psrass23l 20116 psrcom 20117 psrass23 20118 mplcoe1 20174 psropprmul 20334 coe1mul2 20365 gsumfsum 20540 regsumsupp 20694 frlmlbs 20869 tsmsgsum 22674 rrxcph 23922 rrxsuppss 23933 rrxmval 23935 mdegfval 24583 mdegleb 24585 mdegldg 24587 deg1mul3le 24637 wilthlem3 25574 suppovss 30354 fsuppcurry1 30387 fsuppcurry2 30388 fedgmullem1 30924 fdivmpt 44528 fdivmptf 44529 refdivmptf 44530 fdivpm 44531 refdivpm 44532 |
Copyright terms: Public domain | W3C validator |