Theorem dffn4 5275

Description: A function maps onto its range. (Contributed by set.mm contributors, 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 2353	. . 3 ⊢ ran F = ran F
2	1	biantru 491	. 2 ⊢ (F Fn A ↔ (F Fn A ∧ ran F = ran F))
3		df-fo 4793	. 2 ⊢ (F:A–onto→ran F ↔ (F Fn A ∧ ran F = ran F))
4	2, 3	bitr4i 243	1 ⊢ (F Fn A ↔ F:A–onto→ran F)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4773   Fn wfn 4776  –onto→wfo 4779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2346  df-fo 4793

This theorem is referenced by:  funforn  5276  ffoss  5314  mapsn  6026