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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 6746 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
| 4 | 1, 3 | sseqtrd 4020 | 1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 dom cdm 5685 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 df-fn 6564 df-f 6565 |
| This theorem is referenced by: ordtypelem7 9564 vdwlem11 17029 gsumzoppg 19962 taylfvallem1 26398 taylply2 26409 taylply2OLD 26410 taylply 26411 dvtaylp 26412 dvntaylp0 26414 taylthlem1 26415 taylthlem2 26416 taylthlem2OLD 26417 tocyccntz 33164 omssubadd 34302 |
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