Theorem frnd 5341

Description: Deduction form of frn 5340. 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 5340	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3111  ran crn 4599  ⟶wf 5178

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 5186

This theorem is referenced by:  difinfsn  7056  ennnfonelemfun  12287  ennnfonelemf1  12288  tgrest  12710  resttopon  12712  rest0  12720  cnrest2r  12778  cnptoprest2  12781  lmss  12787  txbasval  12808  upxp  12813  uptx  12815  hmeores  12856  unirnblps  12963  unirnbl  12964