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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 6729 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrid 4035 | 1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3949 dom cdm 5677 ⟶wf 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 df-fn 6547 df-f 6548 |
This theorem is referenced by: fisuppfi 9370 wemapso 9546 wemapso2lem 9547 cantnfcl 9662 cantnfle 9666 cantnflt 9667 cantnff 9669 cantnfp1lem3 9675 cantnflem1b 9681 cantnflem1 9684 cantnflem3 9686 cnfcomlem 9694 cnfcom 9695 cnfcom3lem 9698 cnfcom3 9699 fin1a2lem7 10401 isercolllem2 15612 isercolllem3 15613 fsumss 15671 fprodss 15892 vdwlem1 16914 vdwlem5 16918 vdwlem6 16919 ghmpreima 19114 pmtrfconj 19334 gsumval3lem1 19773 gsumval3lem2 19774 gsumval3 19775 gsumzres 19777 gsumzcl2 19778 gsumzf1o 19780 gsumzmhm 19805 gsumzoppg 19812 gsum2d 19840 dpjidcl 19928 lmhmpreima 20659 gsumfsum 21012 regsumsupp 21175 frlmlbs 21352 mplcoe1 21592 mplcoe5 21595 psr1baslem 21709 mdetdiaglem 22100 cnclima 22772 iscncl 22773 cnclsi 22776 txcnmpt 23128 qtopval2 23200 qtopcn 23218 rnelfmlem 23456 fmfnfmlem4 23461 clssubg 23613 tgphaus 23621 tsmsgsum 23643 xmeter 23939 metustss 24060 metustexhalf 24065 restmetu 24079 rrxcph 24909 rrxsuppss 24920 mdegfval 25580 mdegleb 25582 mdegldg 25584 deg1mul3le 25634 plyeq0lem 25724 dgrcl 25747 dgrub 25748 dgrlb 25750 vieta1lem1 25823 suppovss 31906 pwrssmgc 32170 gsumhashmul 32208 rhmpreimaidl 32537 elrspunidl 32546 dimkerim 32712 fedgmullem1 32714 carsggect 33317 sibfof 33339 eulerpartlemsv2 33357 eulerpartlemsf 33358 eulerpartlemt 33370 eulerpartlemgu 33376 eulerpartlemgs2 33379 cvmliftmolem1 34272 cvmlift3lem6 34315 itg2addnclem 36539 keridl 36900 ismrcd1 41436 istopclsd 41438 pwfi2f1o 41838 sge0f1o 45098 smfsuplem1 45527 |
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