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Mirrors > Home > ILE Home > Th. List > f1ff1 | GIF version |
Description: If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.) |
Ref | Expression |
---|---|
f1ff1 | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴⟶𝐶) → 𝐹:𝐴–1-1→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frn 5281 | . 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
2 | f1ssr 5335 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) | |
3 | 1, 2 | sylan2 284 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴⟶𝐶) → 𝐹:𝐴–1-1→𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊆ wss 3071 ran crn 4540 ⟶wf 5119 –1-1→wf1 5120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-f 5127 df-f1 5128 |
This theorem is referenced by: f1resf1 5338 inresflem 6945 |
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