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Mirrors > Home > MPE Home > Th. List > fssdmd | 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, deduction form. (Contributed by AV, 21-Aug-2022.) |
Ref | Expression |
---|---|
fssdmd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fssdmd.d | ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) |
Ref | Expression |
---|---|
fssdmd | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssdmd.d | . 2 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) | |
2 | fssdmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6719 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrd 4015 | 1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3941 dom cdm 5667 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3948 df-ss 3958 df-fn 6537 df-f 6538 |
This theorem is referenced by: ordtypelem7 9516 vdwlem11 16929 gsumzoppg 19860 taylfvallem1 26234 taylply2 26245 taylply 26246 dvtaylp 26247 dvntaylp0 26249 taylthlem1 26250 taylthlem2 26251 tocyccntz 32797 omssubadd 33819 gg-taylthlem2 35684 |
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