Theorem dffn4 5486

Description: A function maps onto its range. (Contributed by NM, 10-May-1998.)

Assertion

Ref	Expression
dffn4	⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)

Proof of Theorem dffn4

Step	Hyp	Ref	Expression
1		eqid 2196	. . 3 ⊢ ran 𝐹 = ran 𝐹
2	1	biantru 302	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹))
3		df-fo 5264	. 2 ⊢ (𝐹:𝐴–onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹))
4	2, 3	bitr4i 187	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ran crn 4664   Fn wfn 5253  –onto→wfo 5256

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-gen 1463  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-fo 5264

This theorem is referenced by:  funforn  5487  fimadmfo  5489  ffoss  5536  tposf2  6326  mapsn  6749  fifo  7046  quslem  12967  gausslemma2dlem1f1o  15301