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| Mirrors > Home > MPE Home > Th. List > fndmu | Structured version Visualization version GIF version | ||
| Description: A function has a unique domain. (Contributed by NM, 11-Aug-1994.) | 
| Ref | Expression | 
|---|---|
| fndmu | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fndm 6671 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 2 | fndm 6671 | . 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵) | |
| 3 | 1, 2 | sylan9req 2798 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 dom cdm 5685 Fn wfn 6556 | 
| 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-fn 6564 | 
| This theorem is referenced by: fodmrnu 6828 0fz1 13584 lmodfopnelem1 20896 grporn 30540 hon0 31812 2ffzoeq 47339 | 
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