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Mirrors > Home > ILE Home > Th. List > f1fn | GIF version |
Description: A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
f1fn | ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 5387 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | ffn 5331 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Fn wfn 5177 ⟶wf 5178 –1-1→wf1 5179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 |
This theorem depends on definitions: df-bi 116 df-f 5186 df-f1 5187 |
This theorem is referenced by: f1fun 5390 f1rel 5391 f1dm 5392 f1ssr 5394 f1f1orn 5437 f1elima 5735 f1eqcocnv 5753 f1oiso 5788 phplem4dom 6819 f1finf1o 6903 updjudhcoinlf 7036 updjudhcoinrg 7037 updjud 7038 fihashf1rn 10691 |
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