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Mirrors > Home > MPE Home > Th. List > ffdm | Structured version Visualization version GIF version |
Description: A mapping is a partial function. (Contributed by NM, 25-Nov-2007.) |
Ref | Expression |
---|---|
ffdm | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 6736 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | 1 | feq2d 6713 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ↔ 𝐹:𝐴⟶𝐵)) |
3 | 2 | ibir 267 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵) |
4 | eqimss 4037 | . . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
6 | 3, 5 | jca 510 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ⊆ wss 3946 dom cdm 5681 ⟶wf 6549 |
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-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-ss 3963 df-fn 6556 df-f 6557 |
This theorem is referenced by: ffdmd 6758 smoiso 8391 s4f1o 14922 islindf2 21804 fourierdlem92 45756 fouriersw 45789 etransclem2 45794 |
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