Theorem frnd 5347

Description: Deduction form of frn 5346. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
frnd.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
frnd	⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)

Proof of Theorem frnd

Step	Hyp	Ref	Expression
1		frnd.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		frn 5346	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3116  ran crn 4605  ⟶wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-f 5192

This theorem is referenced by:  difinfsn  7065  ennnfonelemfun  12350  ennnfonelemf1  12351  tgrest  12809  resttopon  12811  rest0  12819  cnrest2r  12877  cnptoprest2  12880  lmss  12886  txbasval  12907  upxp  12912  uptx  12914  hmeores  12955  unirnblps  13062  unirnbl  13063