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Theorem dffo2 5273
 Description: Alternate definition of an onto function. (Contributed by set.mm contributors, 22-Mar-2006.)
Assertion
Ref Expression
dffo2 (F:AontoB ↔ (F:A–→B ran F = B))

Proof of Theorem dffo2
StepHypRef Expression
1 fof 5269 . . 3 (F:AontoBF:A–→B)
2 forn 5272 . . 3 (F:AontoB → ran F = B)
31, 2jca 518 . 2 (F:AontoB → (F:A–→B ran F = B))
4 ffn 5223 . . 3 (F:A–→BF Fn A)
5 df-fo 4793 . . . 4 (F:AontoB ↔ (F Fn A ran F = B))
65biimpri 197 . . 3 ((F Fn A ran F = B) → F:AontoB)
74, 6sylan 457 . 2 ((F:A–→B ran F = B) → F:AontoB)
83, 7impbii 180 1 (F:AontoB ↔ (F:A–→B ran F = B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 theorem 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-fo 4793 This theorem is referenced by:  foco  5279  foconst  5280  dff1o5  5295  dffo3  5422
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