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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 6652 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | fndm 6652 | . 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵) | |
3 | 1, 2 | sylan9req 2793 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 dom cdm 5676 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-fn 6546 |
This theorem is referenced by: fodmrnu 6813 0fz1 13520 lmodfopnelem1 20507 grporn 29769 hon0 31041 2ffzoeq 46026 |
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