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Mirrors > Home > ILE Home > Th. List > f1of1 | GIF version |
Description: A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.) |
Ref | Expression |
---|---|
f1of1 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 4909 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | 1 | simplbi 259 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 –1-1→wf1 4899 –onto→wfo 4900 –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-f1o 4909 |
This theorem is referenced by: f1of 5126 f1oresrab 5329 f1ocnvfvrneq 5422 isores3 5455 isoini2 5458 f1oiso 5465 f1opw2 5706 tposf12 5884 enssdom 6242 phplem4 6318 phplem4on 6329 fidceq 6330 |
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