Theorem dffn3 5451

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 3217	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2	1	biantru 302	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
3		df-f 5289	. 2 ⊢ (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
4	2, 3	bitr4i 187	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   ∧ wa 104   ↔ wb 105   ⊆ wss 3170  ran crn 4689   Fn wfn 5280  ⟶wf 5281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3176  df-ss 3183  df-f 5289

This theorem is referenced by:  fsn2  5772  fo2ndf  6331