Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6523 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrd 4007 | 1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3936 dom cdm 5555 ⟶wf 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 df-fn 6358 df-f 6359 |
This theorem is referenced by: ordtypelem7 8988 vdwlem11 16327 gsumzoppg 19064 taylfvallem1 24945 taylply2 24956 taylply 24957 dvtaylp 24958 dvntaylp0 24960 taylthlem1 24961 taylthlem2 24962 tocyccntz 30786 omssubadd 31558 |
Copyright terms: Public domain | W3C validator |