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Mirrors > Home > MPE Home > Th. List > fssdm | Structured version Visualization version GIF version |
Description: Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.) |
Ref | Expression |
---|---|
fssdm.d | ⊢ 𝐷 ⊆ dom 𝐹 |
fssdm.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
fssdm | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssdm.d | . 2 ⊢ 𝐷 ⊆ dom 𝐹 | |
2 | fssdm.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6725 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrid 4033 | 1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3947 dom cdm 5675 ⟶wf 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3954 df-ss 3964 df-fn 6543 df-f 6544 |
This theorem is referenced by: fisuppfi 9366 wemapso 9542 wemapso2lem 9543 cantnfcl 9658 cantnfle 9662 cantnflt 9663 cantnff 9665 cantnfp1lem3 9671 cantnflem1b 9677 cantnflem1 9680 cantnflem3 9682 cnfcomlem 9690 cnfcom 9691 cnfcom3lem 9694 cnfcom3 9695 fin1a2lem7 10397 isercolllem2 15608 isercolllem3 15609 fsumss 15667 fprodss 15888 vdwlem1 16910 vdwlem5 16914 vdwlem6 16915 ghmpreima 19108 pmtrfconj 19328 gsumval3lem1 19767 gsumval3lem2 19768 gsumval3 19769 gsumzres 19771 gsumzcl2 19772 gsumzf1o 19774 gsumzmhm 19799 gsumzoppg 19806 gsum2d 19834 dpjidcl 19922 lmhmpreima 20651 gsumfsum 21004 regsumsupp 21166 frlmlbs 21343 mplcoe1 21583 mplcoe5 21586 psr1baslem 21700 mdetdiaglem 22091 cnclima 22763 iscncl 22764 cnclsi 22767 txcnmpt 23119 qtopval2 23191 qtopcn 23209 rnelfmlem 23447 fmfnfmlem4 23452 clssubg 23604 tgphaus 23612 tsmsgsum 23634 xmeter 23930 metustss 24051 metustexhalf 24056 restmetu 24070 rrxcph 24900 rrxsuppss 24911 mdegfval 25571 mdegleb 25573 mdegldg 25575 deg1mul3le 25625 plyeq0lem 25715 dgrcl 25738 dgrub 25739 dgrlb 25741 vieta1lem1 25814 suppovss 31893 pwrssmgc 32157 gsumhashmul 32195 rhmpreimaidl 32525 elrspunidl 32534 dimkerim 32700 fedgmullem1 32702 carsggect 33305 sibfof 33327 eulerpartlemsv2 33345 eulerpartlemsf 33346 eulerpartlemt 33358 eulerpartlemgu 33364 eulerpartlemgs2 33367 cvmliftmolem1 34260 cvmlift3lem6 34303 itg2addnclem 36527 keridl 36888 ismrcd1 41421 istopclsd 41423 pwfi2f1o 41823 sge0f1o 45084 smfsuplem1 45513 |
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