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Mirrors > Home > ILE Home > Th. List > f1ofun | GIF version |
Description: A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.) |
Ref | Expression |
---|---|
f1ofun | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ofn 5127 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fnfun 4996 | . 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Fun wfun 4896 Fn wfn 4897 –1-1-onto→wf1o 4901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 |
This theorem depends on definitions: df-bi 110 df-fn 4905 df-f 4906 df-f1 4907 df-f1o 4909 |
This theorem is referenced by: f1orel 5129 f1oresrab 5329 isose 5460 f1opw 5707 xpcomco 6300 |
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