Theorem f1rn 5602

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 5601	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		frn 5545	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3220  ran crn 4777  ⟶wf 5376  –1-1→wf1 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-f 5384  df-f1 5385

This theorem is used by:  fun11iun  5663  f1dmex  6349  f1finf1o  7269  caserel  7433  djudom  7439  exmidfodomrlemim  7559  hashf1lem2  11308  usgruspgrben  16632  domomsubct  17236