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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 5389 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | fndm 5281 | . 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 1342 dom cdm 4598 Fn wfn 5177 –1-1→wf1 5179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 |
This theorem depends on definitions: df-bi 116 df-fn 5185 df-f 5186 df-f1 5187 |
This theorem is referenced by: fun11iun 5447 tposf12 6228 f1dmvrnfibi 6900 f1vrnfibi 6901 exmidfodomrlemim 7148 hmeoimaf1o 12861 exmidsbthrlem 13742 |
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