Theorem dffn3 5358

Description: A function maps to its range. (Contributed by NM, 1-Sep-1999.)

Assertion

Ref	Expression
dffn3	⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)

Proof of Theorem dffn3

Step	Hyp	Ref	Expression
1		ssid 3167	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2	1	biantru 300	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
3		df-f 5202	. 2 ⊢ (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
4	2, 3	bitr4i 186	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   ∧ wa 103   ↔ wb 104   ⊆ wss 3121  ran crn 4612   Fn wfn 5193  ⟶wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202

This theorem is referenced by:  fsn2  5670  fo2ndf  6206