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Mirrors > Home > ILE Home > Th. List > fssdmd | 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 5368 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrd 3193 | 1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3129 dom cdm 4623 ⟶wf 5208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 df-fn 5215 df-f 5216 |
This theorem is referenced by: (None) |
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