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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 6601 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | fndm 6601 | . 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵) | |
3 | 1, 2 | sylan9req 2799 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 dom cdm 5631 Fn wfn 6487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2730 df-fn 6495 |
This theorem is referenced by: fodmrnu 6760 0fz1 13391 lmodfopnelem1 20287 grporn 29268 hon0 30540 2ffzoeq 45351 |
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