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Mirrors > Home > ILE Home > Th. List > f1dm | GIF version |
Description: The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
f1dm | ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1fn 5253 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | fndm 5147 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 dom cdm 4467 Fn wfn 5044 –1-1→wf1 5046 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 |
This theorem depends on definitions: df-bi 116 df-fn 5052 df-f 5053 df-f1 5054 |
This theorem is referenced by: fun11iun 5309 tposf12 6072 f1dmvrnfibi 6733 f1vrnfibi 6734 exmidfodomrlemim 6924 exmidsbthrlem 12621 |
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