Theorem dffn3 5230

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

Assertion

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

Proof of Theorem dffn3

Step	Hyp	Ref	Expression
1		ssid 3291	. . 3 ⊢ ran F ⊆ ran F
2	1	biantru 491	. 2 ⊢ (F Fn A ↔ (F Fn A ∧ ran F ⊆ ran F))
3		df-f 4792	. 2 ⊢ (F:A–→ran F ↔ (F Fn A ∧ ran F ⊆ ran F))
4	2, 3	bitr4i 243	1 ⊢ (F Fn A ↔ F:A–→ran F)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ⊆ wss 3258  ran crn 4774   Fn wfn 4777  –→wf 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792

This theorem is used by:  fsn2  5435