Theorem f1ff1 5401

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.)

Assertion

Ref	Expression
f1ff1	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴⟶𝐶) → 𝐹:𝐴–1-1→𝐶)

Proof of Theorem f1ff1

Step	Hyp	Ref	Expression
1		frn 5346	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
2		f1ssr 5400	. 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)

Syntax hints:   → wi 4   ∧ wa 103   ⊆ wss 3116  ran crn 4605  ⟶wf 5184  –1-1→wf1 5185

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 5192  df-f1 5193

This theorem is referenced by:  f1resf1  5403  inresflem  7025