Theorem f1rn 5543

Description: The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)

Assertion

Ref	Expression
f1rn	⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)

Proof of Theorem f1rn

Step	Hyp	Ref	Expression
1		f1f 5542	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		frn 5491	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3200  ran crn 4726  ⟶wf 5322  –1-1→wf1 5323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-f 5330  df-f1 5331

This theorem is referenced by:  fun11iun  5604  f1dmex  6277  f1finf1o  7145  caserel  7285  djudom  7291  exmidfodomrlemim  7411  usgruspgrben  16036  domomsubct  16602