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Theorem f1orn 5296
 Description: A one-to-one function maps onto its range. (Contributed by set.mm contributors, 13-Aug-2004.)
Assertion
Ref Expression
f1orn (F:A1-1-onto→ran F ↔ (F Fn A Fun F))

Proof of Theorem f1orn
StepHypRef Expression
1 df-3an 936 . 2 ((F Fn A Fun F ran F = ran F) ↔ ((F Fn A Fun F) ran F = ran F))
2 dff1o2 5291 . 2 (F:A1-1-onto→ran F ↔ (F Fn A Fun F ran F = ran F))
3 eqid 2353 . . 3 ran F = ran F
43biantru 491 . 2 ((F Fn A Fun F) ↔ ((F Fn A Fun F) ran F = ran F))
51, 2, 43bitr4i 268 1 (F:A1-1-onto→ran F ↔ (F Fn A Fun F))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642  ◡ccnv 4771  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –1-1-onto→wf1o 4780 This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  f1f1orn  5297
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